2022 Actions of sl 2 on algebras appearing in categoriﬁcation

We prove that many of the recently-constructed algebras and categories which appear in categoriﬁca-tion can be equipped with an action of sl 2 by derivations. The sl 2 representations which appear are ﬁltered by tensor products of coverma modules. In a future paper, we will address the implications of the sl 2 structure for categoriﬁcation.


Introduction 1.sl 2 structures
Here are several 2-categories with graded 2-morphism spaces which play fundamental roles in categorical representation theory in type A.
They are each defined by generators and relations over some base ring, using the technology of planar diagrammatics.Since the relations only have integral coefficients, we can and will assume the base ring is the integers.The first main theorem of this paper says that these categories have a surprising new structure.
Theorem 1.1.Each of the categories listed above admits an action of sl 2 by derivations, compatible with the graded 2-category structure.Moreover, divided powers of the raising and lowering operators act on the integral form.
More precisely, we will define three operators {d, h, z} on the morphism spaces in these categories, and each operator x ∈ {d, h, z} will satisfy the Leibniz rule with respect to both vertical (•) and horizontal (⊗) composition.The triple (d, h, −z) will act as an sl 2 triple on each morphism space.A graded preadditive category equipped with an action of sl 2 by derivations will be called an sl 2 -category.An sl 2 -category is analogous to a dg-category, but for an unusual kind of homological algebra; morphism spaces in both categories are modules over a Hopf algebra.The concept of an sl 2 -category does not fit precisely into the framework of Hopfological algebra [Kho16,Qi14], but we hope to address this in a follow-up paper.Let us discuss these three operators in turn.
The degree +2 operator d has been the central topic of study in the recent programme which attempts to categorify key objects in representation theory (e.g.quantum groups and Hecke algebras) at a root of unity.For each of the categories above, the operator d has been defined in previous works [KQ15,EQ16a,EQ16b,EQ20].There is a large family of degree +2 derivations one could place on each of these categories, but d is unique (up to duality, see Remark 1.2) in satisfying certain key properties important for categorification.There is still no geometric understanding for the existence and importance of d, though see [BC18] for more on the connection to Steenrod operations.We will not discuss categorification at a root of unity any further in this paper, though we will have much to say in the next paper.

The polynomial ring
If A and B are algebras equipped with an action of sl 2 by derivations, and M and N are bimodules equipped with a compatible action of sl 2 , then the space of bimodule morphisms Hom(M, N ) is naturally equipped with an sl 2 action as well.This is analogous to the internal Hom between two chain complexes, which is itself another chain complex.
There is a well-known action of sl 2 on the ring R n = [x 1 , . . ., x n ], where deg x i = 2 for all i (which determines the h action), and where Here, sl 2 appears as a subalgebra of the Witt Lie algebra acting on polynomials1 .Note that z acts trivially on the subring generated by the roots (x i − x j ), and d does not preserve this subring.For all the categories above, morphism spaces have polynomial subalgebras which play an important role.In all cases, these polynomial subalgebras are preserved by the sl 2 action, and the two different sl 2 triples (d, h, −z) and (d, h, −z) restrict to the same standard sl 2 triple on the polynomial ring.
Most of the categories we are discussing have full faithful embeddings into the category of bimodules over polynomial rings.For example, the Hecke category H is equivalent (after base change) to the category of Soergel bimodules, certain (R n , R n ) bimodules.To give another example, the nilHecke algebra NH n is isomorphic to End R Sn n (R n ).However, we wish to emphasize that this realization in terms of bimodules does not equip NH n or H with an sl 2 action!One must still choose an action of sl 2 on the bimodules in question; this choice is not unique and is rather subtle in practice.Even a free module of rank 1 over R n admits many compatible sl 2 actions.
Example 1.4.In [KQ15,equations (65) and (66) on page 44], a one-parameter family of degree +2 derivations d a is defined on NH n .We define the lowering operator z on NH n below in (1.6b).For any scalar a, (d a , h, −z) is an sl 2 triple.A major point in [KQ15] is that only two of these raising operators, d ±1 , have the desired properties for categorification.The subtle properties of d a are invisible from the perspective of the polynomial ring R n , since (d a , h, −z) restricts to the standard sl 2 triple on R n for all a.
Remark 1.5.In [KR16], Khovanov and Rozansky use the action of the positive half2 of the Witt Lie algebra on R n to place an action of this same algebra on triply graded knot homology, which is built using Hochschild homology of Soergel bimodules.This (positive half of the) Witt action is an important precursor to our sl 2 action, though just as in the previous remark, an action on the polynomial ring does not determine a unique action on the Hecke category itself.
Remark 1.6.The sl 2 action (even on the polynomial ring) currently lacks a geometric motivation.The raising operator is related to general homological operations (e.g.Steenrod squares), though the connection is subtle, see [BC18] (and also see [Kit18] for more on Steenrod operations and Soergel bimodules).However, the lowering operator does not seem to arise from a general construction.Though a good explanation is missing, one thing is clear: sl 2 acts on the * -equivariant cohomology of a point.Geometric constructions of these categories involve perverse sheaves which are equivariant over an algebraic group.We suspect that there is a relationship between the sl 2 action and the existence of a copy of * (i.e.G m ) inside the algebraic group which is acting trivially.Insisting upon equivariance for a trivial action often leads to extra "homological" operations of higher degree, such as the "log of monodromy" maps from [BY13].
For example, the Hecke category studies B × B-equivariant sheaves on G, and any element of Z(G) ∩ B will act the same way on both sides, so its antidiagonal copy in B × B will act trivially.When G = GL n ( ), there is a central copy of * in the torus, and there is also an sl 2 action on the Hecke category.When G = SL n ( ), there is no center, and sl 2 does not act (c.f.[EQ20, Proposition 6.9])!As a shadow of this fact, the reader can already verify that sl 2 acts on [x 1 , . . ., x n ], but it does not have a (nontrivial) S n -invariant action on the subring generated by (x i − x j ) for i < j.This contrasts the B-equivariant cohomology of a point for the Borel subgroup in GL n ( ) versus SL n ( ).
Remark 1.7.There are a number of (typically non-monoidal) categories which also play major roles in categorical representation theory in type A, and for which the operator d has already been studied [KSQ17,QS18].
• The cyclotomic quotients of Khovanov-Lauda-Rouquier categories, which categorify irreducible representations of the quantum group.
• Webster's categories, which categorify tensor products of irreducible representations.
However, neither the cyclotomic quotients nor Webster's categories admit actions of sl 2 .For example, cyclotomic quotients are quotients by an ideal inside U + (gl n ) which is preserved by d but not by z.This suggests some nuance in how one should interpret modules over sl 2 -categories.
Remark 1.8.Other important categories are the cell subquotients of H(S n ), which categorify the irreducible modules over the Hecke algebra, and the Schur quotients of quantum group categorifications [MSV13].They arise from monoidal ideals generated by identity maps of various objects, and identity maps are killed by the sl 2 action.Thus the sl 2 action on H(S n ) descends to its cell subquotients, and the sl 2 action on U(gl 2 ) descends to the Schur quotient.

Contrasting algebra and representation theory
Given an sl 2 -category, we can forget some structure and study it in two ways: • Forgetting the sl 2 action, we can study the category algebraically.We can study the splitting of objects into direct summands, the Jacobson radical, and so forth.
• Forgetting the algebra structure, we can study Hom spaces as representations of sl 2 .We can ask about their characters, their finite-dimensional subrepresentations, and so forth.
There seems to be an incredible connection between the structure of these categories as algebras and as sl 2 representations.It is so astounding to the authors that it needs to be showcased immediately.We hope this example will whet the appetite, and drum up excitement for our sl 2 action.First we present a toy example.For ease of discussion3 let us work over a field k of characteristic zero.The ring k[x] has graded Jacobson radical (x), and the quotient by this ideal is k.So we have a short exact sequence of k[x]-modules, and k is the graded semisimplification of k[x].The submodule (x) is also preserved by the raising operator d, so this is a short exact sequence of U (b + )-modules, where b + is the Lie algebra inside sl 2 generated by d and h.However, (x) is not preserved by z.On the other hand, k ⊂ k[x] is a subalgebra, and is also preserved by the sl 2 action.So we have a short exact sequence of sl 2 -modules, where Q is the quotient module.Note that Q is simple, so it has no finite-dimensional submodules.
The short exact sequences (1.4) and (1.5) live in different categories, but they are both sequences of vector spaces.They split each other, in that the first map of (1.5) will give a section for the quotient map of (1.4).Consequently, we can identify Q with (x) as a vector space and as a complementary direct summand to k.In this fashion, the maximal finite-dimensional sl 2 -submodule k forms a semisimple subalgebra which maps isomorphically to the semisimplification (i.e. the quotient by the Jacobson radical).
Let us reproduce the same behavior in a more interesting example.The nilHecke algebra NH n is the endomorphism algebra of the object E n inside U + (sl 2 ).Put together, the nilHecke algebras form a monoidal category with morphism algebra NH = n≥0 NH n , which is monoidally generated by morphisms depicted as a dot and a crossing.The sl 2 -module structure is defined on the generators below, and is extended to the whole category using the Leibniz rule.
It is well-known that NH n is the endomorphism algebra of the polynomial ring over its subalgebra R Sn of invariant polynomials.The sl 2 -action on R n is S n -equivariant and thus descends to an sl 2 action on R Sn .By the Chevalley theorem, R is free over R Sn of rank n!.Choosing a basis, one obtains an isomorphism between NH n and a matrix algebra (1.7) On [KQ15, page 56] one can find a basis of NH n which corresponds to the basis of matrix entries, and results about the action of d on this basis.
The (graded) Jacobson radical of R Sn is the ideal R Sn + spanned by positive degree elements.The Jacobson radical of NH n is therefore Jac(NH n ) ∼ = Mat n! (R Sn + ). (1.8) Consequently there is a short exact sequence where ss(NH n ) ∼ = Mat n! (k) is the semisimplification of NH n , viewed as a quotient.This is a short exact sequence of NH n -modules.Now let us examine the sl 2 structure on NH n .The ideal R Sn + ⊂ R Sn is preserved by d and h but not by z, since z(e 1 ) = n • id.Consequently, Jac(NH n ) is preserved by d and h but not by z, so there is no induced sl 2 structure on ss(NH n ).The short exact sequence (1.9) is not a short exact sequence of sl 2 representations.However, Mat n! (k) is not just a quotient of NH n ∼ = Mat n! (R Sn ), it is also a subring.See §4.3 for the proof of the following theorem.
Theorem 1.9.The maximal finite-dimensional sl 2 -subrepresentation of NH n is a subalgebra, which maps isomorphically onto the semisimplification of NH n .Moreover, under the identification of NH n with Mat n! (R Sn ) used in [KQ15, Proposition 3.24], this subalgebra is precisely Mat n! (k).
Example 1.10.When n = 2, the following morphisms correspond to the matrix entries in Mat 2 (k).
The reader should confirm that this four-dimensional subspace of NH 2 is preserved by z and d, and is isomorphic as an sl 2 -representation to V ⊗ V * , where V is the standard representation of sl 2 .
Thus one has a short exact sequence of sl 2 -representations where Q is defined as this quotient.This is not a short exact sequence of NH n -modules, since Mat n! (k) is a subring but not an ideal.However, both sequences (1.9) and (1.11) are sequences of k-modules, and split each other.Thus we can identify Q with the Jacobson radical as a vector space complementary to Mat n! (k).Said another way, the finite-dimensional part of the sl 2 -representation NH n is precisely a complement to the Jacobson radical!To summarize, there is a splitting of U (b + )-modules (1.12) When viewed as modules over NH n or its subring R, this is not a splitting but a filtration, with Mat n! (R Sn + ) being the submodule.When viewed as modules over sl 2 , this is not a splitting but a filtration, with Mat n! (k) being the submodule.
Remark 1.11.We come across this situation often in this paper.A vector space (resp.-module) X is isomorphic to a direct sum A ⊕ B ⊕ C as vector spaces.However, when X is equipped with additional structure (e.g. an action of sl 2 , or of R n ), there is no longer a direct sum decomposition.Instead, the subspaces A and A ⊕ B are preserved by the action.In this context we use the slightly imprecise phrasing that the splitting becomes/is a filtration in the presence of the extra structure.We call A the subobject and C the quotient object.The objects A, B, and C were not equipped a priori with any additional structure, and they inherit this structure only by the canonical identification of vector spaces with subquotients in the filtration 0 Note that NH n is infinite-dimensional and even infinitely-generated as an sl 2 module (when n > 1), so this kind of representation does not conform to most familiar regimes (though they are direct limits of representations in category O).Thankfully, NH n has finite-dimensional weight spaces, with weights bounded below.It is easy to prove that such an sl 2 -representation contains a unique maximal finite-dimensional subrepresentation, which we call its core.Equivalently, the core consists of all vectors on which d acts nilpotently.From the Leibniz rule, one can see that the core must be closed under multiplication.Basic facts about the core are proven in §2.The example of NH n is supposed to demonstrate that the core of an sl 2 -category has dramatic significance to the algebraic structure of the category.
One nice feature of the core is that it can be found using basic linear algebra.One need only compute the kernel of d, and then use the lowering operator z to produce the rest.This can be done in each Hom space independently, and does not involve the composition of morphisms.This simplicity is in contrast to other attempts to find complements to the Jacobson radical, by computing inclusion and projection maps to indecomposable summands.This involves much more complicated linear algebra and in-depth knowledge of the category.
We do not wish the reader to expect that the core of an sl 2 -algebra always projects isomorphically to the semisimplification, as this is false in more complicated examples, see §3.4.However, we conjecture that the map from the core to the semisimplification is injective, at least in geometrically-motivated 2-categories.
Conjecture 1.12.In the 2-categories discussed in this paper, the core of each morphism space intersects the Jacobson radical trivially (every morphism in the core is split), so the map from the core to the semisimplification is injective.
For example, one implication of this conjecture is that any degree zero morphism in the kernel of d must split.If p is the projection map to an indecomposable direct summand and d acts nilpotently on p, then every nonzero d k (p) is also a projection map.
Remark 1.13.This conjecture is already quite deep, indicating the non-degeneracy of certain "local intersection forms" upon restriction to finite-dimensional sl 2 representations.One expects the eventual proof of this conjecture to come from a new kind of Hodge-theoretic argument.In all the examples computed to date, local intersection forms have satisfied (an analogue of) the Hodge-Riemann bilinear relations upon restriction to the core.Note that the raising operator d differs from the Lefschetz operator of left multiplication studied in [EW14], and d is not even a Lefschetz operator in the traditional sense.

Filtrations on morphism spaces
Hopefully, we have convinced the reader that the study of these categories as sl 2 -modules, and in particular the study of their finite-dimensional submodules, is of great interest.Morphism spaces in these categories are free modules over a polynomial ring, as well as being modules over sl 2 .This is a useful tool in our effort to understand the sl 2 -module structure on these morphism spaces.
Definition 1.14.The polynomial sl 2 -algebra (R n , sl 2 ) is the polynomial ring R n = [x 1 , . . ., x n ] equipped with the sl 2 action given in (1.3).We also let R n denote the base change of R n to any commutative base ring k.
An (R n , sl 2 )-module is an R n -module M which is also an sl 2 -module, satisfying a Leibniz rule.For x ∈ {d, h, z}, if we write x M for the action on M and x R for the action on R n , the Leibniz rule states that (1.13) Now we ask: what kinds of (R n , sl 2 )-modules appear as morphism spaces in categories of interest?We know that morphism spaces will be free over R n , meaning that they are a direct sum of rank 1 free modules, but one should not expect such a splitting to be respected by the sl 2 structure.The punchline will be: morphism spaces have (R n , sl 2 )-filtrations which are split over U (b − ), whose subquotients are rank 1 free modules over R n .Before stating the result, let us investigate rank 1 free modules, which are easy to classify.Definition 1.15.Let p = a i x i be a linear polynomial in R n , and let Σ(p) ∈ be an integer whose image in k agrees with a i .There is a free rank one graded R n -module R n p with generator 1 p living in degree Σ(p).We define an (R n , sl 2 )-module structure on R n p by setting for any (homogeneous) g ∈ R.
The formulas (1.14) are determined by the Leibniz rule from the action on the generator 1 p : , though Σ is more descriptive notation.In Proposition 2.26 we prove that every (R n , sl 2 )-module which is free of rank one as a graded R n -module is isomorphic to R n p for some p.If p = p ′ then R n p and R n p ′ are non-isomorphic.
Definition 1.16.Let M be an (R n , sl 2 )-module which is free and finitely generated as a graded R n -module.Thus there is a finite set I such that M = i∈I M i as graded R n -modules, and each M i is free of rank 1 over R n .A downfree filtration on M is a splitting into free rank one R n -modules as above, where 1.Each M i is preserved by z.
2. The indexing set I can be equipped with a partial order, so that d(M i ) ⊂ j≤i M j for all i ∈ I.
A homogeneous basis of M as an R n -module is called downfree if it induces a downfree filtration.Definition 1.17.Let M be an (R n , sl 2 )-module equipped with a downfree filtration.In particular, j≤i M j is an (R n , sl 2 )-submodule, and M is equipped with an I-indexed filtration by (R n , sl 2 )-submodules, where the subquotients are free of rank 1 over R n .Each subquotient must be isomorphic as an (R n , sl 2 )-module to R n p i for some unique p i ∈ R n .The multiset of linear polynomials {p i } n i=1 will be called the downfree character of M , with respect to this filtration.
The second main theorem of this paper says that well-known bases of morphism spaces in the categories of interest are actually downfree, and computes their downfree characters.
Example 1.18.The nilHecke algebra NH 2 on two strands is free of rank (1 + q −2 ) as a left R 2 -module, spanned by the identity and the crossing.Since id is killed by d and z, it generates an sl 2 -submodule In the quotient by R • id, d will send the crossing X to −2x 1 X.So the basis {id, X} is downfree, and the downfree character of NH 2 is {0, −2x 1 }.Note that {Σ(0), Σ(−2x 1 )} = {0, −2} which matches the degrees of this graded basis.If instead we had chosen the raising operator d, we would have gotten downfree character {0, −2x 2 }.Note that NH 2 is also free as a right R 2 module, with the same basis.Because the basis is downfree with character {0, −2x 2 }.The right module character for d matches the left module character for d because they are related by duality, which also swaps the left and right action of R n .
Remark 1.19.For the Hecke category, morphism spaces will also be R n -bimodules, but the duality functor will not interchange these actions.A given basis may induce four different characters, based on whether one selects the left or right action of R n , and whether one chooses d or d.
More generally, associated to any element w ∈ S n and any reduced expression of w, one can construct the corresponding diagram in NH n built from crossings, and this element in NH n is independent of the choice of reduced expression.We denote it ψ w ∈ NH n .Then the elements {ψ w } w∈Sn form a basis of NH n as a left or right R n -module, which we call the nilCoxeter basis.It is not hard to compute that Theorem 1.20.The nilCoxeter basis of NH n over R n is a downfree basis, with partial order given by the Bruhat order on S n .This is proven in Theorem 4.5, which also contains an explicit formula for the downfree character, and the generalization to all simply-laced KLR algebras.
Similarly we can study the Hecke category H(S n ).Now R n is the endomorphism ring of the monoidal identity, so all morphism spaces are naturally R n -bimodules.Following ideas of Libedinsky [Lib08], Elias-Williamson in [EW16b] define the double leaves basis, a basis of morphism spaces as left R n -modules, which is indexed by coterminal Bruhat strolls.Let us summarize Theorem 5.25, which contains an explicit formula for the downfree character.
Theorem 1.21.The double leaves basis of morphism spaces in H(S n ) over R n is downfree, with partial order given by the lexicoBruhat order on coterminal Bruhat strolls.
The natural bases of morphism spaces of U(sl 2 ) and U(sl 2 ) are also downfree (conjecturally, since we do not prove it here), though over a different base ring.Any given (nonzero) morphism space in U(sl 2 ) has 2n points on the boundary (n oriented in and n oriented out).This morphism space is free of rank n! over R n ⊗ Λ, where Λ is the ring of symmetric functions acting by bubbles, and R n acts by dots on the inward-oriented strands.We place an sl 2 structure on Λ in §6.2; the lowering operator z depends on the choice of ambient weight.The combinatorics involved in describing the downfree character have not been developed.

What kinds of sl 2 -modules appear?
Having just described the kinds of (R n , sl 2 )-modules which appear in practice, we can ask about what these modules look like as sl 2 -representations, with an eye towards understanding their cores.
In this paper, all sl 2 representations have weights which are bounded below rather than above, so verma modules ∆(k) are defined by inducing from U (b − ) rather than U (b + ).Let L k denote the irreducible weight representation of lowest weight k for all k, a quotient of ∆(k) and submodule of ∇(k).Note that L k is finite-dimensional if and only if k ≤ 0.
Example 1.22.Consider [x] as an sl 2 -module.With its usual basis of monomials, the module looks like this.
. This is a coverma module, with the trivial module (spanned by the identity element) as a submodule.The quotient by the trivial submodule is isomorphic to ∆(2).Note that ∆(2) ∼ = ∇(2) when we work over or in finite characteristic.
As both a ring and as an sl 2 -module, we have [x 1 , . . ., x n ] ∼ = [x] ⊗n .The following proposition is very easy to prove.Proposition 1.23.(See Proposition 2.26) For p = a i x i ∈ R n , there is an isomorphism of sl 2 modules (1.20) In particular, Core(R n p ) is nonzero if and only if a i ∈ ≤0 for all i, in which case is four dimensional, and if p = −3x 1 + x 2 then Core(M ) = 0.This illustrates why the character of an (R n , sl 2 )-module is more useful than the graded degree.
Suppose one has an (R n , sl 2 )-module with a downfree filtration, and one knows the downfree character.By the proposition above, one knows the core of the associated graded module.A priori, this does not make it any easier to determine the core of M , because a finite-dimensional submodule of a subquotient of M need not lead to a finite-dimensional submodule of M itself.However, extensions between (R n , sl 2 )modules are even more limited than extensions between their underlying sl 2 -modules, and sometimes the downfree character of M will determine the core of M !Let us illustrate this with the following result.Even the associated graded has no core unless a ≤ 0, so assume a ≤ 0. There are no extensions between ∇(a) and ∇(a + 2) unless a = 0, by the usual theory of central characters.
Remark 1.26.The lack of extensions between ∇(a) and ∇(a + 2) does not mean that the downfree filtration splits.The splitting as an sl 2 -module and the splitting as an R 1 -module are not compatible.
Remark 1.27.When a = 0 and b = 2, one can find an extension of [x] 0 by [x] 2x with zero core, whereas the core of the associated graded would be one-dimensional.This theorem implies that, for a two-step downfree filtration in one variable, the difference between the core of the original module and its associated graded is at most one copy of the trivial module.Similarly, one can prove that a three-step filtration can remove a copy of L 0 or L −1 , but not L −k for k ≥ 2.
Remark 1.28.The situation is more complex in more than one variable, because polynomials in the roots (x i − x j ) are killed by z, and this allows for more extensions (such polynomials times 1 b are valid choices for m in (1.22)).See §5.2 for an example.Interestingly, many of these extensions do not admit integrallydefined divided powers!Keeping track of divided powers and integrality properties does seem to rigidify the possible extensions.
The (R n , sl 2 )-modules admitting a downfree filtration form a reasonably nice category which we feel is important to study.We hope to provide a methodical study of (R n , sl 2 )-modules in future work, and provide only the basics in this paper.

Conclusion
In §2 we provide some basic results and definitions related to sl 2 -categories and their modules, and the special case of polynomial rings.In the subsequent sections we examine the categories U + (g) (in simply laced type), H, U(sl 2 ), and U(sl 2 ) in turn, constructing the derivation z, establishing the sl 2 action, and verifying the claims made in this introduction about downfree filtrations.In §6.1 and §6.2 we discuss the sl 2 action on symmetric polynomials and symmetric functions, which may be of independent interest.
In §5, in order to prove results about the downfree filtration on the Hecke category, we need to establish some basic properties of light leaves and double leaves.Aside from this, the proofs in this paper are all relatively straightforward computations.
We find this new sl 2 structure to be extremely tantalizing.In a follow-up paper we will introduce an sl 2 -enrichment of the categories of interest, where multiplicity spaces are naturally finite-dimensional sl 2 representations.We also introduce a conjectural approach to defining a triangulated category from an sl 2 -category, analogous to the derived categories associated to p-dg algebras in [Kho16,Qi14], whose Grothendieck group is naturally a module over [q + q −1 ], the Grothendieck ring of finite-dimensional sl 2 representations.We will state a number of conjectures about cores and Jacobson radicals, which translate into the existence of nice bases for the triangulated Grothendieck group.If true, these conjectures would produce a new kind of categorification and a new kind of canonical basis, where structure coefficients are naturally unimodal, being the graded dimensions of sl 2 representations.For example, the Grothendieck group of the diagrammatic Hecke category would be a variant of the Hecke algebra with base ring [q + q −1 ].Our conjectures would also imply several open conjectures about categorification at a root of unity, e.g. that the p-dg Grothendieck group of the diagrammatic Hecke category is the Hecke algebra at a root of unity.
While some aspects of the material developed in the follow-up paper are quite technical and lengthy to explain, other aspects are accessible and directly computational.We continue our introduction in §3 by explaining the basics of the theory of sl 2 decompositions.In particular, this discussion further motivates the study of the core, and of iterated cores that one obtains by combining the algebra structure and the sl 2 structure on an sl 2 -category.We discuss examples in the diagrammatic Hecke category in §3.4, and state our new sl 2 -adapted Soergel categorification conjecture in §3.5.
Remark 1.29.This final remark is for those readers familiar with p-dg algebras and categorification at a root of unity.By forgetting the lowering operator and restricting from U (sl 2 ) to U (b + ), one obtains a p-dg structure on these categories.The biggest problem in computing the p-dg Grothenieck group is to prove that any object has a fantastic filtration whose subquotients are certain p-dg-indecomposable objects.Historically this has been done by computing idempotent decompositions explicitly, but this method becomes intractable quickly.
Our sl 2 -adapted Soergel categorification conjecture essentially states that Bott-Samelson objects have sl 2fantastic filtrations, whose subquotients are "indecomposable objects" tensored with a multiplicity space which is a finite-dimensional sl 2 representation.Because a finite-dimensional sl 2 representation has a filtration by one-dimensional U (b + )-modules, this will yield a filtration by p-dg indecomposable objects (with multiplicity one); an sl 2 -fantastic filtration will be a p-dg fantastic filtration.However, sl 2 -fantastic filtrations are more restrictive and include more structure, which ironically makes them easier to find.Once you compute the highest weight vector, you can apply z to find a basis for the rest of the representation; this was a tool which was not previously available.

Representations of polynomial sl -algebras
Most of the results in this chapter are relatively straightforward, but because we are not aware of any literature on the topic, we provide some details.

g-algebras and Leibniz exercises
Definition 2.1.Let k be a commutative domain, and let g be a Lie algebra over k.A g-algebra is a k-algebra A equipped with an action of g by derivations.We sometimes write (A, g) for this structure.One can define a g-category similarly, and g will act on each Hom space.A monoidal g-category is a g-category with the additional requirement for each x ∈ g that (2.1a) By the interchange law, it is equivalent to require that By default in this paper k = .Before continuing, let us address the practical question of what it takes to place a g-algebra structure on A.Here are two simplifying lemmas.The first reduces the data required to define a single derivation.Lemma 2.2.Let A be a k-algebra given by generators and relations.To define a derivation x on A, it suffices to specify x(a) for each generator of A, and to check that x preserves all the relations of A (what this means precisely is stated in the proof).
Proof.Let S be the generating set of A, and F S the free algebra on these generators.Any assignment x : S → F S will extend to a unique derivation on F S , using the Leibniz rule to define the action of x on a word in S. If I is the ideal in F S generated by the relations, and x sends the generating relations to elements of I, then x preserves I by the Leibniz rule.Thus x descends to a derivation on A.
The second lemma reduces the work required to check that a collection of derivations gives an action of a particular Lie algebra g.Lemma 2.3.Let x, y, z be three derivations on an algebra A, and suppose that (2.2) holds for a generating set of elements a ∈ A. Then it holds for all a ∈ A.
Proof.The equation (2.2) is clearly linear in a, so it remains to check that if (2.2) holds for a and b, then it holds for the product ab.We compute Here is one more practical consideration.
Definition 2.4.Let x be a derivation on a -algebra A. The divided powers of x are the operators which a priori send A to A⊗ É.We say that the divided powers x (k) are defined over k or just defined integrally4 if there are operators x (k) defined in A (without any base change) such that (2.5) Lemma 2.5.Let x be a derivation on a -algebra A, and suppose that x (k) (a) is defined integrally for a generating set of elements in A. Then x (k) is defined integrally (on all of A).
Proof.Again, x (k) is a linear operator.It is an easy exercise in the Leibniz rule (and the binomial theorem) that In particular, if x (k) is defined on a and b for all k, then it is defined on ab.
We write ⊗ for ⊗ k .
Corollary 2.6.Let A and B be g-algebras.Then the tensor product algebra A ⊗ B, with its tensor product g-action, is a g-algebra.If the divided powers of x ∈ g are defined integrally on A and on B, then they are defined integrally on A ⊗ B.
Proof.Since A⊗B is generated by elements of A and elements of B, Lemmas 2.3 and 2.5 imply the result.
Remark 2.7.The g-action on an algebra A can be more conceptually understood in the language of a module-algebra over a Hopf algebra H (see, e.g., [Mon93] for more details).Let H be a Hopf algebra over k.We will always assume that H is a projective k-module.The comultiplication on H is denoted ∆ in Sweedler's notation: for any h ∈ H, and the counit map is denoted ǫ : H → k.A k-algebra A is called an H-module algebra if A is an H-module, and the multiplication map of A is compatible with the H-action: where 1 A is the identity element of A. When H and A are graded (super)algebras, the notions should be adapted so that the H action on A respects the graded (super)algebra structures.As a particular case, for a g-algebra A, one may take H to be the universal enveloping algebra of a Lie algebra g over k, with the H-action on A induced by derivations of g on A. Similarly, taking to be the graded superalgebra of dual numbers, where the degree of d is set to be 1 and one recovers the usual notion of a differential graded algebra as an H-module algebra.

Modules over g-algebras
Definition 2.8.Let (A, g) be a g-algebra.An (A, g)-module is an A-module M which is equipped with a g action.For x ∈ g write x M for the action of x on M .We require the compatibility for any a ∈ A and m ∈ M .A morphism of (A, g)-modules is an A-module morphism M → N which intertwines the action of g.These form a space denoted Hom (A,g) (M, N ).
This definition is completely analogous to the definition of a dg-module over a dg-algebra.Instead of just one differential, we keep track of a g action.Just as for dg-modules, we can consider the internal Hom space.
Definition 2.9.Let M and N be two (A, g)-modules.Then the space of A-module maps Hom A (M, N ) can be equipped with the structure of a g-module, where (2.11) Proposition 2.10.If A is commutative, then Hom A (M, N ) is also an A-module.The actions of A and g are compatible, making Hom A (M, N ) into an (A, g)-module.
Proof.We only need to check the Leibniz rule.We compute (2.12) The result follows.
Lemma 2.11.Let M be an A-module with a presentation.To give M the structure of an (A, g)-module, it suffices to define x M (m) for all generators m ∈ M and all x ∈ g, and to check the relations.For a given x ∈ g, the divided powers of x M are defined integrally if and only if they are defined integrally on the generators of M .
Proof.This is entirely analogous to Lemmas 2.2, 2.3, and 2.5.We leave the proof to the reader.
Remark 2.12.Let A be an H-module algebra as in Remark 2.7, one may form the smash product ring A#H, which, as an algebra, is isomorphic to the tensor product algebra A ⊗ k H, whose multiplication is given by for any a, b ∈ A and h, k ∈ H.The abelian category of A#H-modules is acted upon by the abelian monoidal category of k-projective H-representations.Let us denote this action by ⊠.Given an A#H-module M and a k-projective By construction, (-) ⊠ V is an exact functor on A#H-modules.
In this language, an (A, g)-module is no other than a module over A#H, where H is the universal enveloping algebra of g over k.It follows readily that the category of (A, g)-modules constitute an abelian category, with a categorical module structure over the monoidal category of projective H-representations.

Weights
When g = sl 2 we change notation to impose one additional condition: that h acts semisimply.Definition 2.13.A (weight) sl 2 -algebra (over k) is a k-algebra A equipped with an action of sl 2 by derivations, on which the Cartan element h ∈ sl 2 acts diagonalizably (with eigenvalues in k).A (weight) (A, sl 2 )-module is an A-module M with a compatible sl 2 action, on which the Cartan element h ∈ sl 2 acts diagonalizably.A divided powers sl 2 -algebra is an sl 2 -algebra where the divided powers of d and z are integrally defined, and similarly for a divided powers (A, sl 2 )-module.
We let (d, h, −z) be the standard sl 2 triple, and we write x A or x M for the action of x on A or M , where x ∈ {d, h, z}.
Lemma 2.14.Equip A (resp.M ) with a k-grading by the eigenvalues of h, the usual weight grading.Then A is a graded algebra (resp.and M is a graded module).
Proof.For homogeneous elements a 1 and m 2 of weights d 1 and d 2 respectively, the Leibniz rule for h implies that a 1 m 2 has weight Most often in this paper the eigenvalues of h will be integers, and our rings and modules will begraded.Only in this chapter on generalities, and to some extent in §6.2, will we care about more general weights.
Remark 2.15.For all the examples we study in this paper, the eigenvalues live in the image of the map → k.Even when this map is not injective, all our examples can be compatibly -graded.In this context a few statements need to be modified in the obvious way.For example, for an element a ∈ k we will often write "a ∈ " to indicate that a is in the image of → k, but in the -graded context one should instead choose a preimage a ∈ .Without further ado, we assume that our examples are -graded even in finite characteristic.
Let us make one remark about divided powers.Definition 2.16.Let U = U (sl 2 ) denote the idempotented divided powers form of the enveloping algebra of sl 2 .A definition can be found in, for instance, [KLMS12, Section 3.1] (by specializing the q to be 1 in the quantum setting).All modules over U will be assumed to be weight modules, so that the idempotent 1 n acts by multiplication by n for n ∈ .
Note that U acts on any sl 2 -module where divided powers exist and the weights are integers.An action of U on A and B extends to an action on A ⊗ B (Corollary 2.6) since U is a Hopf algebra.The proof that U is a Hopf algebra is essentially the same as the proof of Lemma 2.5.A divided powers sl 2 -algebra over is the same thing as an algebra in the category of (weight) U -modules.

On representations of sl 2 over the integers
The reader may be familiar with the properties of category O, but many things are different and slightly unfamiliar when working over the integers.The goal of this section is to make precise what we mean by Verma and coVerma modules, and to warn the reader of some pitfalls.Because we study z instead of −z, and bounded-below modules rather than bounded-above modules, some signs may differ from expectations.
Definition 2.17.For k ∈ k we will define modules ∆(k) and ∇(k) over sl 2 as follows.

∆(k) :
• Let us rephrase these pictures in formulas.The module ∆(k) has a free -basis {v k,m } m≥0 , corresponding to the dots labeled by m in the picture above, where v k,m has weight k + 2m.Set v k,m := 0 for m < 0. We have Similarly, ∇(k) has a free -basis {w k,m }, where w k,m has weight k + 2m.Set w k,m := 0 for m < 0. We have (2.17) Lemma 2.18.When a ∈ , divided powers are defined integrally on ∆(a) and ∇(a).More generally, divided powers are defined in k whenever the a-binomial coefficients are defined in k for all l ∈ and m ∈ ≥0 : these are the elements Proof.It is a simple computation that For example, suppose k = [y] and a = y.Then is not an element of k, so divided powers are not defined integrally on ∆(a) or ∇(a).
For the rest of this section, k = , so we can think of Verma and coVerma modules as U -modules.Let v k := v k,0 and w k := w k,0 .Note that (2.20) The module ∆(k) is generated by v k over U , and the divided powers d (m) applied to this generator give the basis.Meanwhile, ∇(k) is infinitely-generated over U , but it is "co-generated" by w k , as the divided powers z (m) bring every basis element to w k .Given any U -module M which is free as a -module, 2M ⊂ M will be a proper submodule.Simplicity, in its naive sense as when working over a field, is not as useful a concept.Definition 2.19.A morphism of U -modules is called h-split if it is a split morphism of -modules for each weight space.Let U -spmod denote the category whose objects are U -modules where weight spaces are free over , and whose morphisms are h-split maps.If M is a module in U -spmod, a U -submodule is called h-split if the inclusion map is h-split, and similarly for quotient modules.
Note that any isomorphism, or more generally any genuinely split map (over U ), is automatically hsplit.However, U -spmod is not an additive category, as the sum of h-split morphisms need not be h-split.After all, id + id = 2id.Nonetheless, if f is any h-split morphism then its kernel and cokernel naturally live in U -spmod, so U -spmod shares some features with an abelian category.
When restricting to h-split morphisms, verma and coverma modules have features which resemble those in the familiar category O.
Proposition 2.20.The modules ∆(k) and ∇(k) are indecomposable for all k ∈ .When k > 0, the modules ∆(k) and ∇(k) have no h-split submodules or quotients.There are h-split short exact sequences for all k ≤ 0 given by where W (k) and W ∨ (k), the so-called Weyl and dual Weyl modules, are defined by these short exact sequences.Aside from those given in (2.21), there are no other h-split submodules or quotients of ∆(k) and ∇(k) for k ≤ 0.
Proof.Since all weight spaces are free of rank 1 over , an h-split submodule is determined by which weight spaces it includes.Now the usual arguments (analyzing which arrows have zero coefficient) imply that only certain collections of weight spaces can give a submodule.One need only verify that when k ≤ 0 the submodule of ∆(k) generated in degree −k + 2 is ∇(−k + 2), which one can do directly from the pictures (2.15) or the corresponding formulas.
What is more interesting is the following observation.
Note a major difference between this setting and the usual category O in characteristic zero.Normally, ∆(k) ∼ = ∇(k) for all k > 0, and one reverses the placement of ∆(−k + 2) and ∇(−k + 2) in (2.21); they are isomorphic, so it doesn't matter, but it is often done to help illustrate the verma and coverma resolutions of simple modules.Over , one is not permitted to swap ∆(−k + 2) and ∇(−k + 2), and the Weyl and dual Weyl modules do not have Verma or coVerma resolutions.This fact will be of great importance in the sequel.
Proof.Clearly ∇(1) ∼ = ∆(1) by sending w 1,m → v 1,m , since the formulas (2.16) and (2.17) agree.When k ≤ 0 clearly ∇(k) ∼ = ∆(k) since one has a finite rank submodule and the other does not.So suppose k > 0. What happens over É is well-known, so assume that there is some a ≥ 1 such that {1, . . ., a} is invertible in k and a + 1 is not invertible.We claim that the image of z (a) in degree k is different when comparing ∇(k) to ∆(k), which implies they are non-isomorphic.Clearly w k is in the image of z (a) inside ∇(k), whereas only the span of (a + 1)v k is in the image of z (a) inside ∆(k).
Remark 2.22.If M is any module with a lowest weight vector v in weight k, then there is a morphism ∆(k) → M sending v k → v.This morphism will not necessarily be h-split.For example, the natural morphism ∆(k) → ∇(k) is not h-split except when k = 1.
Proof.If we send w 0,m → x m , we can confirm (2.17) easily.
Remark 2.24.One is tempted to say that the ideal (x) is isomorphic to ∆(2), but one must be careful with this statement.Ideals are usually thought of as submodules, so that (x) is identified with a subset of R 1 .This subset is an R 1 -submodule but not a U -submodule, because it is not preserved by z.This was discussed at more length in the introduction, see the toy example from §1.3.Definition 2.25.Let a ∈ k and let R 1 a denote the free rank one graded R 1 module with generator 1 a .We give it an (R 1 , sl 2 )-module structure by setting and extending these operators to all of R 1 a by the Leibniz rule.
That R 1 a is well-defined is a consequence of the following proposition.
(i) Any (R 1 , sl 2 )-module structure on the rank-one free module R 1 is isomorphic to R 1 (a) for a unique a ∈ k.
Proof.For a rank-one module, denote by v a generator as an R 1 -module.Then, the module is also generated by v as an (R 1 , sl 2 )-module.For degree reasons, there must be a, a ′ ∈ k such that The commutation relation [d, −z] = h applied to the generator v shows that For the second statement, if we send w a,m → x m • 1 a , we can confirm (2.17) easily.Now we let R n = k[x 1 , . . ., x n ] be the polynomial sl 2 -algebra, as in Definition 1.14.
Lemma 2.27.As an algebra and an sl 2 -module we have In particular, divided powers are well-defined.
Proof.The isomorphism with the tensor product is obvious.See Corollary 2.6 for the rest.
We recall a definition from the introduction.
Definition 2.28.Let p = a i x i be a linear polynomial in R n , and let There is a free rank one graded R n -module R n p with generator 1 p .We define an (R n , sl 2 )-module structure on R n p by setting and extending by the Leibniz rule.
Proposition 2.29.As an sl 2 -module, (2.24) In particular, divided powers are well-defined when a i ∈ for all i.
Proof.It is easily verified that the map Proposition 2.30.Every (R n , sl 2 )-module which is free of rank one as a graded R n -module is isomorphic to R n p for some p.
Proof.Let M be such a module, and name the generator 1 M .Then z(1 M ) = 0, and d(1 M ) = p • 1 M for some linear polynomial p, both for degree reasons.The remaining structure follows from the Leibniz rule.The fact that the degree of 1 M must be Σ(p) follows from the fact that

The core
We will eventually be interested in settings where the base ring k is or É or [y] or of finite characteristic.
Thus we are careful in this chapter to make general statements.
Definition 2.32.An sl 2 (k) module is called bounded if • it is a weight module, with weights in , • the set of weights with non-zero weight spaces is bounded below, • and each non-zero weight space is free of finite rank over k.
Proposition 2.33.Suppose that → k is injective and k is Noetherian.Any bounded sl 2 representation M will have a maximal submodule which is finitely generated over k, which we call the core of M and denote Core(M ).It satisfies (2.26) Note that the core may be zero.
Proof.For the sake of this proof, define Core(M ) using (2.26).Note that Core(M ) is a weight module.
Clearly d acts nilpotently on any submodule which is finitely generated as a k-module, so Core(M ) contains all such submodules.We need only prove that Core(M ) is finitely generated over k.
We now argue that Core(M ) is locally finitely generated, i.e. any element is contained in an sl 2 submodule which is finitely generated over k.Suppose that m ∈ M homogeneous is acted upon nilpotently by d.By the PBW theorem, the span of {z a h b d c • m} a,b,c≥0 is an sl 2 subrepresentation containing m.Only finitely many pairs (a, c) will give a nonzero result, and the span of {z a h b d c • m} b≥0 agrees with the span of the single vector z a d c • m.Thus m is contained in a subrepresentation which is finite rank over k.
If Core(M ) is not finitely generated over k, then by the Noetherian hypothesis it must have nonzero elements in infinitely many weight spaces (lest it be contained in a finite rank k-module).Because Core(M ) is locally finitely generated, it must have highest weight vectors in arbitrarily high weights.We now argue that Core(M ) also has (lowest weight) vectors in arbitrary low weights, a contradiction because M is bounded.This last argument is standard, we are just being careful because we have placed very few assumptions on k.
Suppose that Core(M ) has a highest weight vector v in weight k > 0. Let ∆ ′ (k) be the boundedabove Verma module with highest weight k.Then v induces a nonzero map of sl 2 (k)-modules φ : ∆ ′ (k) → Core(M ).Since the target is a submodule of a free k-module it is torsion free over k.Since each weight space in ∆ ′ (k) is free of rank 1 over k, the kernel of φ is divisible, so it is spanned by a subset of the weight spaces in ∆ ′ (k).Now standard arguments imply that the kernel of φ is zero in weight −k, so Core(M ) is nonzero in weight −k.
Remark 2.34.When k is a PID, one can use Smith normal form to deduce that the kernel of a map between free (finite rank) modules is a split summand of the source.One can use this to prove that Core(M ) is a h-split submodule of M .We imagine this is always true (even when k is a Dedekind domain or something unusual) but we do not have the knowledge or examples to say one way or another.

Lemma 2.35.
There is a left-exact functor Core from bounded sl 2 -modules to sl 2 -modules which are finitely generated over k, sending M to Core(M ), and restricting any morphism to the core.
Proof.The image under an sl 2 -intertwiner of a finitely generated module is a finitely generated submodule.Thus the core is sent to the core under any sl 2 -intertwiner.Injective maps restrict to injective maps, so Core is left-exact.
Remark 2.36.Note that Core is not right exact.For example, for a ≤ 0 the canonical map ∆(a) → W (a) is surjective, but Core(∆(a)) = 0 while Core(W (a)) = W (a). In this sense, the core of a quotient can be larger than the core of the original module.Of course, the core of a quotient can be smaller too, as the quotient map could kill part of the core. (2.27) Proof.We use the identification (2.24) of R n p with a tensor product of covermas.
Let M be any sl 2 module and consider ∇(a) ⊗ M .Any vector v in this tensor product can be written uniquely as j≥0 w a,j ⊗ m j (2.28) for some m j ∈ M , with finitely many m j being nonzero.Let J be the maximal value of j for which m j = 0.
But by the formula from (2.19), d N (w a,J ) is never zero when a > 0. In fact, it is never zero when J > −a.
As a consequence we deduce that ∇(a) ⊗ M has zero core when a > 0. Similar arguments prove that ∆(a) ⊗ M has no core for any a ∈ .This also follows from the familiar idea that tensoring a verma module with anything yields a module with a verma filtration, and modules with a verma filtration have no finite-dimensional submodules.
This proves that Core(R n p ) = 0 if any a i > 0. If a i ≤ 0 for all i, then the tensor product of the submodules W ∨ ai clearly lives inside Core(R n p ).Since ∇(a i )/W ∨ ai ∼ = ∆(−a i + 2), the quotient of R n p by the tensor product of coweyl modules has a filtration whose subquotients are isomorphic to M ⊗ ∆(a) for some a.If d has no nilpotents on the associated graded of a filtered module, it has no nilpotents on the whole module.This proves that the core of R n p is not bigger than expected.
Corollary 2.38.Let p = a i x i .Then the elements form a basis for the core of R n p .
Proof.This follows from the case n = 1, where it is straightforward.
Finally, we mention one more useful result.The same argument will show that the core of a tensor product contains the tensor product of the cores.Proposition 2.39.In an sl 2 -category where all morphism spaces are bounded sl 2 representations, the core is a subcategory.In a monoidal sl 2 -category with this property, the core is a monoidal subcategory.
Proof.We need to prove that the core contains all identity maps (which is obvious) and is closed under composition (both horizontal and vertical).We use the description of the core as those elements on which

On decompositions in the presence of sl 2
In this quasi-introductory chapter we discuss the nature of direct sum decompositions in the presence of an sl 2 action, and the possible implications for the Grothendieck group.Afterwards, we will reconnect these ideas to the concept of the core.For simplicity, we work with categories over a field k rather than the integers.
The idea has already arisen in categorification at a root of unity (which studies categories with a derivation d) to study a direct sum decomposition not with the traditional use of idempotents, but by studying filtrations on representable functors instead.This is a guiding principle below.

Actions on partial idempotent completions
It is traditional when studying (pre)additive categories over a field to take (the additive closure and then) the Karoubi envelope.One obtains a Krull-Schmidt category, whose split Grothendieck group is spanned by isomorphism classes of indecomposable objects.Then one uses direct sum decompositions to find relations in the Grothendieck group.One way to motivate the Karoubi envelope is that it naturally appears when studying modules over the category: representable modules Hom(X, −) are projective, and thus so are their direct summands Hom(X, −) • e for any idempotent e ∈ End(X).
However, for a category with an sl 2 action, there is no induced action of sl 2 on the Karoubi envelope!Let X be an object and e ∈ End(X) an idempotent.The subspace is not closed under sl 2 in general.For each x ∈ {d, z, h} one can try to define an operator x on Hom(X, −)•e by truncating the usual action, to wit Each operator x satisfies the Leibniz rule.However, these operators will not necessarily satisfy the sl 2 relations.
An sl 2 quotient idempotent instead satisfies x(e)e = 0.An object which has no nontrivial sl 2 submodule idempotents is called sl 2 -indecomposable.
By applying the Leibniz rule to x(e 2 ) = x(e), one deduces that e is an sl 2 submodule idempotent if and only if x(e) = x(e)e.Hence Hom(X, −) • e is closed under x, and the formula (3.1) is another way to write the restriction of the sl 2 action to this submodule.
It is also easy to deduce that e is an sl 2 submodule idempotent if and only if (1 − e) is an sl 2 quotient idempotent.In this case, Hom(X, −) • (1 − e) is not closed under the sl 2 action.However, using the identification Hom(X, −) • (1 − e) ∼ = Hom(X, −)/Hom(X, −) • e, the sl 2 action induced on this quotient agrees with the formulas from (3.1).Let e be an sl 2 submodule idempotent.Looking at representable functors out of the partial idempotent completion which adjoins the images of e and 1 − e, we have a short exact sequence of sl 2 representations which splits as modules for the underlying additive category.In this way, the direct sum decomposition X ∼ = Im(e) ⊕ Im(1 − e) in the additive category becomes a filtration for sl 2 .We still refer to the idempotents e and (1 − e) as complements.
Definition 3.2.An sl 2 decomposition of an object X is an ordered sequence of idempotents e 1 , e 2 , . . ., e n such that 1.The set {e 1 , . . ., e n } is a complete collection of orthogonal idempotents: e i e j = e j e i = 0 for i = j, and n i=1 e i = id X .2. For any k ≤ n, the idempotent e 1 + e 2 + . . .+ e k is an sl 2 submodule idempotent.
In this case, the idempotents e i are called sl 2 subquotient idempotents.
Similarly, the formula (3.1) defines an sl 2 action on Hom(X, −) • e for any sl 2 subquotient idempotent.For the analogous definition in the context of p-dg algebras, see [EQ16b,§4].

Future goals, and cofibrance
The goal is to find a setting where certain idempotent decompositions give rise to relations on the Grothendieck group, much as they did in the split Grothendieck group of the underlying additive category.Another goal is for this Grothendieck group to be naturally a module over the Grothendieck ring of finite-dimensional sl 2 representations.Not every idempotent decomposition should give rise to a relation.For a general idempotent, the subspaces Hom(X, −) • e and Hom(X, −) • (1 − e) are entangled by the sl 2 action, and no filtration (3.2) is present; one does not expect a relation in the Grothendieck group.However, even for submodule idempotents, one does not always expect a relation, because of a technical point we now address.
The program of categorification at a root of unity, which accounts for the action of d but not z, has an established framework for Grothendieck groups like the one we seek.There is a triangulated category associated to a category with an action of d, constructed using Hopfological algebra [Kho16,Qi14].In this triangulated category, the short exact sequence (3.2) does not always give rise to a distinguished triangle, but it does when the module Hom(X, −) • e is cofibrant.In particular, the appropriate category to study is not the partial idempotent completion which adjoins all sl 2 subquotient idempotents, but only those which are cofibrant.
There are typically two ways to prove that the image of an idempotent is cofibrant.The first is to show that it is isomorphic to a representable module, which happens when the idempotent factors compatibly with d: The second method, the two-out-of-three principle, uses the fact that if two out of three terms in (3.2) are cofibrant, then so is the third.Thus the complement of a representable module in a representable module is cofibrant.Adjoining its image, one can assume it is representable for future decompositions.
So to summarize, one expects relations on the Grothendieck group coming from cofibrant short exact sequences like (3.2), and cofibrance can be confirmed by factoring the idempotent as a projection and inclusion, and checking d(p) • i = 0.For a longer discussion of these thorny issues, see [EQ16b,§4].
Everything below in this section should be taken with a grain of salt.The authors' expectation is that there is a triangulated category one can build from any category with sl 2 action, where short exact sequences coming from cofibrant sl 2 submodule idempotents give rise to distinguished triangles.If one can factor an idempotent as e = i • p, where i and p have degree zero, then it will be cofibrant if x(p) • i = 0 for all x ∈ sl 2 .However, there are additional cofibrant objects coming from representable modules in the enriched category.
Definition 3.3.Let C be an sl 2 -category.The sl 2 -enriched category is the sl 2 -category C enrich defined as follows.Its objects are formal tensor products B ⊠ V (see Remark 2.12 for an explanation of the notation), for an object B of C and a finite-dimensional representation V of sl 2 .We set as an sl 2 -module.Composition is given by There is a monoidal action of the category of finite-dimensional sl 2 representations on C enrich , where on objects one has One should think about B ⊠ V as though it were the direct sum of dim(V ) copies of B in C (with the appropriate grading shifts).Then Hom C enrich (B ⊠ V, B ′ ⊠ V ′ ) can be viewed appropriately as matrices of morphisms between B and B ′ .However, the action of sl 2 on V and V ′ is used to twist the action on these matrices of morphisms.If dim(V ) > 1 then B ⊠ V is decomposable as an object in the underlying category C, but can be sl 2 -indecomposable when V is indecomposable.
We develop the enriched category more formally in the sequel.This method of enrichment is similar to techniques one encounters when studying the equivariantization-deequivariantization principle, see e.g.[DGNO10, §4].In the next section we discuss a practical method to find an sl 2 submodule idempotent whose image is B ⊠ V , generalizing the equations in (3.3).
Representable modules from the enriched category should also be considered as cofibrant.Taking the sl 2 -enriched category is analogous to taking the additive closure of a preadditive category, and adjoining the images of cofibrant sl 2 subquotient idempotents is analogous to taking the Karoubi envelope.Both are operations one typically does before attempting to take a Grothendieck group.Below, we refer to the partial idempotent completion which adds cofibrant sl 2 subquotient idempotents as the cofibrant Karoubi envelope.
At the moment there are only conjectural ideas for how to build a triangulated category from an sl 2enriched category with an appropriate Grothendieck group.These ideas (sl 2 -omological algebra) will be explored in the sequel.However, one can explicitly state conjectures for the expected behavior of the 2categories studied in this paper, which would be required for this triangulated category to be well-behaved, such as our sl 2 -adapted Soergel categorification conjecture in §3.5 below.

Decompositions and the core
For sanity, throughout this section we work over a field of characteristic zero.Recall the divided powers operators x (k) := x k k! for k ≥ 1.It may help the reader to read this section concurrently with the following one, which gives examples of the general phenomena discussed here.
Suppose we have objects X and B in an sl 2 category, and we wish to construct an sl 2 submodule idempotent whose image is B ⊠ V where, for example, V is the three-dimensional irreducible module for sl 2 .In the underlying additive category, we would need to prove that which we would do by constructing inclusion and projection maps.Specifically, we would find maps p j : X → B of degree j for j ∈ {−2, 0, 2}, and inclusion maps i j of degree −j, satisfying We set e j = i j • p j and e = e j .Suppose can find maps p −2 and i 2 of degree −2, satisfying the equations Then d(p 2 ) = 0, and {p −2 , p 0 , p 2 } is the standard basis for V (styled as a Weyl module) sitting isomorphically inside the core of Hom(X, B).Similarly one can define However, we make no assumption that d 3 (i 2 ) = 0! Were this true, then {i 2 , −i 0 , i −2 } would be the dual standard basis for V (styled as a dual Weyl module) sitting inside Hom(B, X).In practice, Hom(B, X) may have no finite-dimensional sl 2 subrepresentations.Applying d to the equality p 2 i 2 = id B we obtain so p 2 i 0 = 0. Similarly, applying d and z repeatedly we can deduce that (3.5) holds.Moreover, let φ = d(i −2 ), which measures the failure of the span of {i 2 , i 0 , i −2 } to be a finite-dimensional subrepresentation.Again using the Leibniz rule, we deduce that for all j ∈ {−2, 0, 2}.
We leave the reader to verify that In particular, e 2 is a d-submodule idempotent and a z-quotient idempotent, but not an sl 2 subquotient idempotent.Similarly, e −2 is a d-quotient idempotent and a z-submodule idempotent, and e 0 is a subquotient for both, but neither of these are sl 2 subquotient idempotents.However, recalling that e = e j , we have ed(e) = 0, ez(e) = 0.
Thus e is an sl 2 submodule idempotent.We claim that Hom(X, −) • e is a submodule of Hom(X, −) which is isomorphic as an sl 2 -module to Hom(B, −) ⊠ V .
Remark 3.4.The inclusion and projection maps i j and p j form an Fc-filtration on B ⊠ V for d in the sense of [EQ16b,§4.7].They also form an Fc-filtration for z, with the opposite order.
The reader may imagine how this generalizes to other irreducible representations of sl 2 .Finding projection and inclusion maps of negative-most degree which satisfy the analogue of (3.6) is our primary tool for constructing cofibrant sl 2 -submodule idempotents.As one can see, this is intimately related with finding the core of Hom(X, B), and proving that the maps within are split.
Remark 3.5.It is not obvious that every sl 2 -submodule idempotent of X whose image is isomorphic to B ⊠ V should be constructible in this fashion, and this is a slightly thorny and technical issue, not suitable for this brief overview.We explore this question in the sequel.
Remark 3.6.That projection and inclusion maps of negative-most degree are in the kernel of z is a relatively ubiquitous phenomenon.Maps of negative-most degree are constructed from diagrams without polynomials, which by the formulas in this paper are in the kernel of z.Also, maps of negative-most degree often split when this degree is non-positive (for the diagrammatic Hecke category in characteristic zero, this is related to the Soergel conjecture, and further to hard Lefschetz on multiplicity spaces, see [EW16a]).
Remark 3.7.One can define a Lefschetz form on degree −k elements of Hom(X, B), which sends (f, g) → f • d k (ḡ), where ḡ is the vertical flip of g.This form is valued in End(B).If B is absolutely indecomposable then End(B) is local with maximal ideal m and quotient field End(B)/m ∼ = k.In this case one can define a Lefschetz form valued in k.
In the prototypical example above, when i 2 = p −2 , then the assumption that i 2 pairs nontrivially against d 2 (p −2 ) is the nondegeneracy of a Lefschetz form.In the examples we have computed, this Lefschetz form satisfies the Hodge-Riemann bilinear relations.Thus the existence of cofibrant sl 2 subquotient idempotents is related to the Hodge theory of the operator d, when restricted to the core of Hom(X, B).
The entire discussion above relates to proving that an sl 2 subquotient idempotent has the appropriate factorization behavior, which implies it is cofibrant.What about the other sl 2 subquotient idempotents in an sl 2 decomposition?
If we wish to continue decomposing X, we need to examine the complement of B ⊠ V .If it is not sl 2indecomposable, we seek a cofibrant sl 2 submodule summand isomorphic to B ′ ⊠ V ′ in this complement, which would be an sl 2 subquotient summand of X.To apply our techniques above one does not necessarily need the core of Hom(X, B ′ ) to be nontrivial; one needs the core of Hom(Im(1 − e), B ′ ) to be nontrivial instead.In other words, we want a projection map p ′ j : X → B ′ of non-positive degree for which z(p ′ j ) = 0, and while d −j+1 (p ′ j ) might not be zero, it must be zero modulo Hom(X, −) • e.These secondary projection maps p ′ j live in the core relative to e, while e was generated by projection maps in a genuine core.Thus these projection maps live in an iterated core: the core of some morphism space, considered modulo an sl 2 -invariant ideal generated by elements in the core of some other morphism space (and so forth).
In the sequel we will carefully define the notion of an sl 2 -antastic filtration, which is effectively an sl 2 decomposition where each subquotient idempotent e i is constructed using projection maps which are in the core modulo the ideal generated by e 1 +. ..+e i−1 .For an sl 2 -antastic filtration, the subquotient idempotents e i have cofibrant images.We also permit sl 2 -antastic filtrations where all but one of the idempotents factors as above.The remaining idempotent will be cofibrant by the two-out-of-three principle.

Behavior in examples
To elaborate on the ideas of the previous section, and on the relationship between the core and the semisimplification, we discuss several miracles which occur in the diagrammatic Hecke category H(S n ), and try to avoid distracting the reader with the details.All these examples were computed explicitly in [EQ20], and we give more precise references for each example.Only the raising operator d was studied in [EQ20], but adding z into the mix is relatively easy (often it is zero for degree reasons).We work over a field k.We begin with an example where everything is as easy as possible, studied in [EQ20, §4.3 and 4.4].

Example 3.8.
There is an object B s (for any simple reflection s) satisfying Since the endomorphism ring of B s is a local ring with quotient field k, the semisimplification of End(B s B s ) is isomorphic to Mat 2 (k).The degree −1 projection map p 1 : B s B s → B s (−1) is uniquely determined up to scalar, and the miracle is that d 2 (p 1 ) = 0.
In close analogy to Example 1.10, the core of the sl 2 action on End(B s B s ) is a subalgebra isomorphic to Mat 2 (k), which is isomorphic to V ⊗ V * as an sl 2 representation.The matrix entries have the form i j p k for j, k ∈ {1, 2}, where i j and p k are particular inclusion and projection maps for the decomposition (3.10).Moreover, the core of Hom(B s B s , B s ) is precisely the span of p 1 and p 2 , with d(p 1 ) = p 2 and d(p 2 ) = 0. Similarly, the core of Hom(B s , B s B s ) is the span of i 1 and i 2 with d(i 2 ) = i 1 and d(i 1 ) = 0.
Note however that neither primitive idempotent e 1 = i 1 p 1 or e 2 = i 2 p 2 = 1 − e 1 is an sl 2 submodule idempotent.We have e 1 z(e 1 ) = 0 but e 1 d(e 1 ) = 0, and vice versa for e 2 .Thus B s B s is sl 2 -indecomposable.
In the enriched category, there is an isomorphism where k 2 is the standard representation of sl 2 .
The next example was done in great detail in [EQ20, §6.4 and 6.5, clarified further in §6.8].
Example 3.9.In the Hecke category of S 3 , there is a specific object5 X which splits as a direct sum of two non-isomorphic indecomposable objects Y and Z.Let J denote the Jacobson radical of the category.It is the case that End(Y )/J ∼ = k and End(Z)/J ∼ = k, so that End(X)/J is two-dimensional, spanned by the two idempotents which project to these summands.However, the core of End(X) is one-dimensional, spanned by the identity map, and neither primitive idempotent is killed by d.Hence the splitting of (3.11) as modules over H becomes a filtration with respect to the sl 2 action, with Hom(Y, −) • p Y being the submodule.Moreover, each layer of the filtration is generated by its core modulo the ideal generated by the previous part of the filtration.
In contrast, if i Y is the corresponding inclusion map in Hom(Y, X), we have d(i Y ) = 0.This explains why the corresponding idempotent e Y = i Y p Y is not killed by d.However we have the miracles The sl 2 -adapted Soergel categorification conjecture 25 Working with contravariant representable functors instead, we see that the decomposition of functors is an filtration for sl 2 , where this time i Z • Hom(Z, −) is the submodule.Another consequence of these miracles is that e Y d(e Y ) = 0, so e Y is an sl 2 subquotient idempotent with complement e Z .This equips X with an sl 2 -antastic filtration.
In the above example, one could not find sufficiently many projection maps in the core to split X into indecomposable direct summands.However, one could construct iterated cores by taking the core, quotienting by the ideal it generates, taking the core of what remains, quotienting by the ideal it generates, and so forth.One could find sufficiently many projection maps for X within the iterated cores.Similar to Conjecture 1.12, we conjecture that iterated cores do not intersect the Jacobson radical (in the 2-categories we consider in this paper).A thorough development of iterated cores and a formal statement of this conjecture will be found in the sequel to this paper.
Iterated cores are still insufficient to split any object into indecomposable direct summands, as seen in the next example, which is studied in [EQ20, §9].
Example 3.10.In the Hecke category of S 8 , there is an object6 X which splits as a direct sum of two nonisomorphic indecomposable objects Y and Z.The object Z is technically not in the category but in the Karoubi envelope.However, there is no splitting of Hom(X, −) or Hom(−, X) into direct summands as a left module over H which is filtered with respect to sl 2 ; neither e Y nor e Z is an sl 2 subquotient idempotent.These two additive summands are entangled by the sl 2 action.At the same time, the cores of both Hom(X, Y ) and Hom(Y, X) are zero.We can not even define an sl 2 structure on Hom(Z, −) or Hom(−, Z).
In the example above, the sl 2 structure can not be used to find a complement for the Jacobson radical, and X is sl 2 -indecomposable.This may be a feature rather than a bug!In characteristic 2 the splitting X ∼ = Y ⊕ Z does not hold, and X is indecomposable; perhaps this is detected by the sl 2 action.
Conjecture 3.11.Let us work in the sl 2 -enriched category of H(S n ).There are objects D y in the cofibrant Karoubi envelope, which satisfy and are inductively defined by the following properties.
1. Let w be a reduced expression for an element w ∈ S n .The Bott-Samelson object BS(w) has an sl 2antastic filtration, and all but one subquotient in the filtration is isomorphic to D y ⊠ V for some finitedimensional sl 2 representation V and some y < w in S n .The remaining subquotient is therefore cofibrant.This last subquotient is sl 2 -indecomposable and not isomorphic to D y ⊠ V for any y < w and any V , and we denote it by D w .
2. The morphisms known as rex moves produce isomorphisms between D w and D w ′ for any two reduced expressions for the same element w.We let D w denote an object in this isomorphism class.
3. Any Bott-Samelson object BS(x), whether x is reduced or not, has a sl 2 -antastic filtration whose subquotients are isomorphic to D y ⊠ V for various finite-dimensional V and various y ∈ S n .
The ordinary Soergel categorification theorem can be stated in almost identical fashion, except using ordinary direct sum decompositions (with grading shifts) rather than sl 2 -antastic filtrations.This is used to define the indecomposable Soergel bimodule B w for w ∈ S n .Note that D w need not be isomorphic to B w in the underlying additive category.In example 3.10, letting w = 35246135724635 which is short for s 3 s 5 s 2 • • • , the Bott-Samelson bimodule X = BS(w) is sl 2 -indecomposable, so D w = X.However, D w is not indecomposable, and its top indecomposable summand B w was called Z above.
If this conjecture holds then, in some conjectural triangulated category, we expect that the objects D w will descend to a basis in the triangulated Grothendieck group, as a free module over the Grothendieck ring of finite-dimensional sl 2 representations.Because an sl 2 -antastic filtration descends to an Fc-filtration for d, we will obtain an analogous result for the p-dg Hecke category, as a free module over the cyclotomic ring [ζ] where ζ 2 is a primitive p-th root of unity.
The Grothendieck ring of this triangulated category should provide a [q + q −1 ]-form of the Hecke algebra.The structure coefficients for the basis [D w ] are unimodal polynomials, because they are the graded dimensions of finite-dimensional sl 2 representations.This does not imply the unimodality of Kazhdan-Lusztig structure coefficients, since D w = B w .
We think of Conjecture 3.11 as stating that the Hecke category "has enough" sl 2 -antastic filtrations, or that it "has enough" split maps in the iterated cores.When trying to generalize this conjecture to the other 2-categories studied in this paper, it is not easy to rigorously state what it means to "have enough," without pinning down implicitly some suspected collection of objects, like {D w }, which generate the category.We have failed to find a concise way to discuss the generalizations of Conjecture 3.11, so we leave further discussion to the next paper where these ideas are developed more formally.For the reader familiar with KLR algebras: a KLR algebra is determined by the polynomials Q ij in the double crossing relation, for each pair of vertices i and j in the (oriented) Dynkin diagram.If i and j are not connected by an edge, Q ij = 0 as usual.If an edge is oriented from i to j, then Q ij = x i − x j .In our action of sl 2 below, we will have for all7 i.It is crucial here that z(Q ij ) = 0, so one can not use the alternative choice One extends the derivation to all diagrams using the monoidal Leibniz rule.
Theorem 4.4.The action of sl 2 on U + (g) given in Definition 4.3 is well-defined.The divided power operators d (k) and z (k) are well-defined in the integral form for all k ≥ 0, making U + (g) into a divided powers sl 2 -algebra.
Proof.First we need to check that the derivations d and z are well-defined.By Lemma 2.2, we need only check that they satisfy the relations of U + (g).(4.7) The sign is determined by the orientation of the edge connecting these two colors, and will not affect the rest of the computation.Applying z to the LHS yields zero.The RHS is Hence z is a well-defined derivation.Now we wish to check that (d, h, −z) is an sl 2 triple, and also that divided powers are well-defined integrally.By Lemmas 2.3 and 2.5, we need only check this on the generators.
Certain Hom spaces are clearly identified as (R, sl 2 )-modules for various polynomial rings R. For example, when i and j are adjacent colors, Hom(E i E j , E j E i ) ∼ = [x i , x j ] as left [x i , x j ]-modules.By construction z kills the generator of the free rank 1 module, and d multiplies it by x j , so the result is precisely the module [x i , x j ] x j .(We should also confirm that the generator lives in degree +1 = Σ(x j ).)We have already proven in Proposition 2.29 that this is a well-defined sl 2 -representation with integrally-defined divided powers.Here are several such isomorphisms (here i and j are adjacent colors, and i and k are distant): This takes care of all generators but the crossing in End(E i E i ).
It is easy to verify that as desired.It is easy to verify that One can see this also from the matrix of (1.10).In particular, when acting on the (i, i) crossing, d (2) is well-defined integrally and d (k) = 0 for all k ≥ 3.

Action on the basis of crossings
It was proven by in [KL09, Theorem 2.5] that each morphism space in U + (g) is a free left (or right) module over the polynomial subring generated by dots, and that a basis can be constructed using diagrams built entirely out of crossings.More precisely, let i = (i 1 , . . ., i n ) and j = (j 1 , . . ., j n ) be sequences of colors, both having length n.Let (S n ) j i denote the subset of S n consisting of those permutations w for which i k = j w(k) for all 1 ≤ k ≤ n.This subset is empty unless the number of strands of each color agrees in i and j.If this happens, and there are a m strands with the color m, then restricting w ∈ (S n ) j i to the strands of color m we get a permutation in S am .This induces a bijection between (S n ) j i and S a1 × • • • × S a d , and this bijection preserves the Bruhat order.In each case, the Bruhat order is generated by the operation which removes a single crossing between two same-colored strands, when the result is still a reduced expression.
Pick a reduced expression of w ∈ (S n ) j i , and let ψ w denote the crossing diagram for that reduced expression, with the strands colored to represent a morphism in Hom(E i , E j ).Then, as a left module over R j (dots acting on the target sequence j), or as a right module over R i (dots acting on the source sequence i), we have {ψ w } w∈(Sn) j i is a basis for Hom(E i , E j ).(4.12) Theorem 4.5.For any w ∈ (S n ) (4.13) Thus {ψ w } induces a downfree filtration of Hom(E i , E j ), parametrized by (S n ) j i with its Bruhat order.In the associated graded, the span of ψ w is a copy of R n p(w) , where a formula for p(w) is given in (4.15) or (4.17), depending on whether we study the left R j -action or the right R i -action.
Proof.Let us first prove the statement in the context of left modules over R j .Clearly z kills ψ w as desired.The result is easiest to see using a different formula for the action of d on same-colored crossings, namely We also recall When we apply d to ψ w , we take the sum of d applied to each crossing: up to linear combinations, this will either add a dot to the northwest of a crossing, or remove the crossing.The first term on the RHS of (4.14) says to remove a same-colored crossing.This produces an expression for an element v < w which is smaller in the Bruhat order.Note that this may not be a reduced expression for v, and even if it is, it may not be the chosen reduced expression ψ v , so relations (such as (4.7)) must be applied to rewrite this as a linear combination of basis diagrams.This rewriting process may produce diagrams which are even lower in the Bruhat order (c.f. the adjacent-colored Reidemeister III relation [KL09, (2.8)]), but it is well-known that the relations of the KLR algebra can be used to simplify an arbitrary diagram to a basis diagram without ever going upwards in the Bruhat order.We state this as Lemma 4.7 below.So whenever we remove a crossing, we get something in the R j -span of ψ v ′ for v ′ < w in the Bruhat order.
Now suppose a dot is placed to the northwest of a crossing.This dot should be forced to the top of the diagram (using (4.3) or (4.5)).On its way up, (4.3) can produce an error term where a same-colored crossing is removed; once again, this will be in the span of diagrams which are smaller in the Bruhat order.
To see what polynomial remains on the top, we need only count the number of times a dot appeared on each strand.
For the k-th strand on top (colored j k ), we count the ways a dot can be added to that strand.We get coefficient −2 each time that strand is the northwest strand of a same-colored crossing, and +1 each time that strand is the northwest strand of an adjacent-colored crossing.Note that the k-th strand on top will cross the ℓ-th strand on top, and be the northwest strand, if and only if k < ℓ and w −1 (ℓ) < w −1 (k).Consequently, let (4.15a) Hence p(w) represents the action of d on the associated graded in the downfree filtration.We may write p(w) more succinctly using the dot product on the set of colors (see [KL09, top of p3]) by the formula Suppose we instead study this morphism space as a right R i -module.Now it helps to use the following formulas instead.
(4.16)The k-th strand on bottom receives a dot whenever it is the southeast strand in a crossing, and if it crosses the ℓ-th strand on bottom, then ℓ < k and w(ℓ) > w(k).So let Then this p(w) describes the associated graded as a right R i -module.
Remark 4.6.Applying the anti-involution which flips each diagram upside-down, we get the analogous result for the sl 2 -triple (d, h, −z).This will swap the left and right action, so it will also swap (4.17) and (4.15).
In the proof above, we used the following statement.
Lemma 4.7.Let D be a crossing diagram corresponding to an subexpression of a reduced expression for w ∈ S n , where we remove one crossing.Then D is in the left R j -span (or the right R i -span) of {ψ v } v≤w .
This lemma is considered obvious by most people in the field (including us), for which reason it is difficult to cite.The result is stated without proof in [KL09, (2.33)], for example.It would not be hard to prove this lemma directly, but it would be several annoying pages without much payoff.We will continue the tradition and not prove the lemma.One could also deduce the result from the main theorem in [Eli19].
While Theorem 4.5 gives control over the associated graded, it does not describe the upper-triangular terms in d explicitly.This was because rewriting non-reduced expressions or the wrong reduced expression in terms of the basis can be complicated.However, for the nilHecke algebra it is easy: nonreduced expressions are zero, and all reduced expressions for the same element are equal.For the nilHecke algebra it is not hard to give a precise formula for d(ψ w ).
Theorem 4.8.Inside NH n we have (4.18) Here, v < 1 w means that v < w in the Bruhat order and ℓ(v) = ℓ(w) − 1.The integers m v,w will be described in the proof.
Note that we study the left action of R n in this proof.We leave the adaptations for the right action to the reader.Proof.To any element w ∈ S n one can associate its inversion set, the set of pairs (a, b) with 1 ≤ a < b ≤ n such that w(a) > w(b).Each crossing in a reduced expression for w produces a single inversion (a, b), and we refer to it as the (a, b)-crossing.In other words, if (a, b) is in the inversion set of w then the ath strand and the bth strand (on bottom) will eventually cross exactly once, and this crossing can be identified with the pair (a, b).Note that b will be the strand on the northwest of the (a, b) crossing; we always write our inversions (a, b) in order so that a < b.
Pick a triple a < b < c and consider the ordered set {(a, b) < (a, c) < (b, c)}, called the packet of the triple (a, b, c).A quick examination (try to draw it) should convince the reader that the inversion set of w intersected with this packet is either a prefix or a suffix.Moreover, for any reduced expression for w, the crossings in a packet either appear in lexicographic order (if a prefix) or antilexicographic order (if a suffix).Only when the entire packet is contained in the inversion set of w can they appear in either lexicographic or antilexicographic order (because the full set is both a prefix and a suffix), and the braid relation sts = tst swaps lexicographic for antilexicographic.For example, if (b, c) is not an inversion, but (a, c) is, then (a, b) must also be an inversion, and (a, b) must occur below (a, c) in a reduced expression for w.These ideas are the start of Manin-Schechtmann's theory of higher Bruhat orders, see [MS89].
If v < 1 w then the inversion set of v is equal to that of w with exactly one inversion removed.Let (a, b) be this inversion.Then (with regards to the left action of R n ) we set In this formula, the condition that w(c) > w(a) is equivalent to the condition that w(c) > w(b).After all, if w(b) < w(c) < w(a) then the packet of (a, c, b) will be entirely contained in the inversion set of w, and removing (a, b) will not yield either a prefix or a suffix, so it will not yield a reduced expression.We now argue that (4.18) holds.Applying d to ψ w as in the proof of Theorem 4.5, we get a sum of diagrams where either a dot is added or a crossing is removed.We have already computed that when all dots reach the top, the overall polynomial is p(w).We need to compute the coefficient with which the (a, b) crossing is removed, for some (a, b) in the inversion set of w.Any crossing which occurs lower down in ψ w , and for which the a or b strand is the northwest-southeast strand, will produce a dot which will eventually be forced through the

The nilHecke algebra and the matrix algebra
We wish to justify some of the claims made in the introduction.For sake of sanity we work in characteristic zero.Let n ≥ 1.
Most aspects of this theorem were proven in [KQ15, Proposition 3.24 and preceding].We give three proofs, mostly for pedagogical reasons.The example of n = 3 is done explicitly after the proofs, and it may help the reader to look at the proofs and the example simultaneously.
Proof.Let V denote the cohomology ring of the flag variety, thought of as the quotient of R n by the ideal generated by positive degree elements of R Sn n .Then V is an n!-dimensional graded vector space, equipped with a natural action of d.By [KL09, Example 2.2.3] there is a ring isomorphism NH n ∼ = Mat n! (R Sn n ), and hence a vector space isomorphism n , where we identify Mat n! (k) with End k (V ).It was proven in [KQ15, Proposition 3.24] that the action of d respects this tensor product decomposition.From this we can immediately deduce that d acts nilpotently on Mat n! (k) ⊂ NH n , so this subalgebra is contained in the core of NH n .
There are a number of ways to confirm that the core of NH n is not bigger than Mat n! (k).Using the tensor decomposition NH n ∼ = End(V ) ⊗ R Sn n , one can verify that nothing else is acted on nilpotently by d.Alternatively, the core of a (bounded below) sl 2 -algebra is a subalgebra by Proposition 2.39, and any subalgebra of Mat n! (R Sn n ) which properly contains Mat n! (k) is infinite-dimensional.Note that V is not naturally an sl 2 -representation: sl 2 acts on R n but does not preserve the ideal generated by positive degree elements of R Sn n , so it does not act on the quotient.However, we know that the core is some finite-dimensional sl 2 -representation, so its isomorphism class is determined by its graded dimension.
Here is a second, more computational proof.
As an sl 2 representation, Proposition 2.37 states that which is a representation of dimension n!.The same is true for R n −δ ′ where δ ′ = 1x 2 +2x 3 +. ..+(n−1)x n , though it may be nicer to order the tensor products in a fashion respecting the indices on the polynomials: In fact, a basis for Core(R n −δ ) is also a basis for R n as a free module over R Sn n .From Schubert theory, it is well-known that the polynomials form a basis for R n over R Sn n .Meanwhile, by Proposition 2.37 and Corollary 2.38, they also form a basis for the core of R n −δ .In similar fashion, for Core(R n −δ ′ ) we can use the basis There is a map of R n -bimodules To make this a map of graded R n -bimodules, we need to shift the source so that 1 ⊗ 1 lives in degree −n(n − 1).Keeping track of the action of d, we get an sl 2 -intertwiner Here we think of R n ⊗ k R n as a polynomial ring in 2n variables, the left variables and the right variables, where δ l uses the left variables and δ ′ r uses the right variables.This map φ is known to be surjective, see e.g. the matrix basis of NH n described in [KLMS12, Proposition 2.16] or [KQ15, Proposition 3.3].It is not injective, because it factors through the quotient R n ⊗ R Sn n R n , though Ker φ must be an sl 2 -submodule.In fact, by counting graded dimensions, one can verify that is an isomorphism.Again by Proposition 2.37 and Corollary 2.38, we know that as sl 2 -representations, being spanned by ⊗ ′ .However, we are interested in the core of the quotient R n , and the core of a quotient module can be both bigger and smaller than the original, see Remark 2.36.Thankfully, we also know that ⊗ ′ is a basis of R n ⊗ R Sn n R n as an R Sn n module, and goes to a basis of NH n over R Sn n , see again [KLMS12, Proposition 2.16].So the map φ from (4.25) is injective on the core.
As in the other proof, once one knows that the core of NH n contains ⊗ ′ or Mat n! (k), there are a number of ways to confirm that it is not bigger.Remark 4.10.Note that ⊗ ′ is not quite the matrix basis of Mat n! (k) ⊂ Mat n! (R Sn n ), though it has the same span.To get the matrix basis one must use dual bases for R n over R Sn n , such as the Schubert and dual Schubert bases.
Here is a sketch of a third proof that the core contains End É (L 0 ⊗ L 1 ⊗ • • • ⊗ L n−1 ).We ignore the remaining parts of the theorem.
Proof.(Sketch) By restricting from R n to R Sn n , R n −δ is an (R Sn n , sl 2 )-module of rank n! with the sl 2 -stable basis .As was shown in [KQ15, Section 3.1] (see the discussion there around equations (63) and (64)), the isomorphism equips the nilHecke algebra with an sl 2 -action, which agrees with Definition 4.3.
We have already argued in §2.6 that the core of a tensor product contains the tensor product of the cores.Since endomorphism rings are particular kinds of tensor products (though one must be careful when taking tensor products of infinite-dimensional representations in this way), one deduces that the core of an endomorphism ring contains the endomorphism ring of the cores.Thus Core(NH n ) contains Example 4.11.Let n = 3.Here is a basis of Core(R 3 −2x 1 − x 2 ), making it clear the isomorphism with L 2 ⊗ L 1 .
Here is another basis making clear the isomorphism with L 3 ⊕ L 1 .
For those interested in integral structure, this second basis is not a -basis, as the determinant of the change of basis matrix is 3.
To obtain analogous bases of R 3 −x 2 − 2x 3 just swap x 1 and x 3 .Now NH 3 is isomorphic to a 6×6 matrix algebra over R S3 3 , and the matrix entries correspond to ψ w0 with certain polynomials on top and on bottom, see e.g.[KQ15, Proposition 3.3].The polynomials on top and those on bottom must be dual bases for R 3 over R S3 3 with respect to the Demazure operator ∂ w0 .However, one can get a basis over R S3 3 (not necessarily a matrix basis) but choosing any two bases for R 3 over R S3 3 , and placing them on top and bottom of ψ w0 .Choosing your favorite bases for Core(R 3 −2x 1 − x 2 ) and Core(R 3 −x 2 − 2x 3 ) respectively for the top and bottom, the sl 2 structure is transparent.Definition 5.2.Equip H with an sl 2 action as follows.The derivation d is the derivation defined in [EQ20, Theorem 2.5], where g i = x i .The derivation z kills all diagrams without polynomials, and sends x i to 1.In the formulas below, blue represents the simple reflection s i = (i, i + 1).Red represents s i+1 , and green represents some color distant from blue. ) (5.1i) One extends the derivation to all diagrams using the monoidal Leibniz rule.
Theorem 5.3.The action of sl 2 on H given in Definition 5.2 is well-defined.The divided power operators d (k) and z (k) are defined integrally for all k ≥ 0, making H into a divided powers sl 2 -algebra.
Proof.As in the proof of Theorem 4.4, we will use the lemmas of §2.1 to reduce the amount of work we need to do.This time we do it tacitly.First we need to check that the derivations d and z preserve the relations of H.That d preserves the relations was checked in [EQ20, Theorems 2.1, 2.3].Let us check that z preserves the relations.
Any diagram without polynomials is sent to zero by z.Any relation which is a linear combination of diagrams without polynomials is therefore preserved by z.There is only one relation that remains, the polynomial forcing relation.Below, blue represents s i , and j is arbitrary. . (5.2) Note that ∂ i (x j ) is a scalar.This relation is preserved by z since z(x i ) = 1 for all i, so z kills the LHS, and the RHS is a diagram without polynomials so is also killed by z.Hence z is a well-defined derivation.Now we wish to check that (d, h, −z) is an sl 2 triple, and also that divided powers are defined integrally.We can check these properties on the generators.Since d and z raise or lower the degree appropriately, we need only check that [z, d] = h.For each of the non-polynomial generators φ, z(φ) = 0, and it is very easy to confirm that z(d(φ)) = (deg φ) • φ = h(φ).
(5.3)Meanwhile, for End(½) = R n , the action of sl 2 is the standard one.This confirms that the sl 2 action is welldefined.That d (k) is defined integrally was checked in [EQ20,§8].That z (k) is defined integrally on the polynomial ring End(½) was checked in Lemma 2.27.That z (k) is defined integrally on the other generators is easy, since it is zero for k ≥ 1.

Reminders: rex moves and lower terms
Definition 5.4.For w ∈ S n , let I <w denote the two-sided ideal spanned by all morphisms which factor through reduced expressions for elements v ∈ S n with v < w.
Given two reduced expressions w, w ′ for the same element w ∈ S n , Matsumoto's theorem states that they can be connected by a sequence of braid relations.To such a sequence of braid relations there is a corresponding diagram built from 4-valent and 6-valent vertices, having source w and target w ′ , which we call a rex move.There are many potential sequences of braid relations which go from w to w ′ , and the corresponding rex moves in H are not equal.
Lemma 5.5.Let w, w ′ be reduced expressions for w ∈ S n .For any two rex moves w → w ′ , their difference lies in I <w .
Rex moves are always in the kernel of z, as any diagram is, but are not always in the kernel of d, see (5.1e).Two different rex moves with the same source and target can have different values of d.For example, let s = s i and t = s i+1 , and consider the reduced expression (s, t, s).The identity map of this reduced expression is a rex move which is killed by d.Meanwhile, the path (s, t, s) → (t, s, t) → (s, t, s) gives the doubled 6-valent vertex, which is not in the kernel of d.The reader versed with this diagrammatic calculus should have no trouble verifying that .
(5.4) Note at least that d sends the doubled 6-valent vertex to the ideal I <sts of lower terms.
Remark 5.6.The right hand side of (5.4) is a left R n -linear combination of double leaves, and double leaves form a basis for morphisms as a left (or right) R n -module.Note the non-trivial root α t which appears.This gives an example of the kind of behavior discussed in Remark 1.28.

Reminders: subexpressions and light leaves
Let x = (s x1 , . . ., s x d ) be an expression of length d, and e ⊂ x be a subexpression.To e we associate a Bruhat stroll (1 = w 0 , w 1 , . . ., w d ) as in [EW16b,§2.4],where to get from w i−1 to w i we multiply by either s xi or by 1 depending on e.Note that the Bruhat stroll determines the subexpression, and vice versa.We refer to w d as the terminus of e.We let E(x, w) denote the set of subexpressions e ⊂ x with terminus w.Following ideas of Libedinsky [Lib08], to each e ∈ E(x, w) we associate in [EW16b, Construction 6.1] a morphism LL e : x → w called a light leaf, whose target is some reduced expression for w (depending on e).Flipping this light leaf upside-down, we get a morphism ΓΓ e : w → x.
Let us remark on some important features of the light leaf construction.If v is a reduced expression for some v ∈ S n , then it has a unique subexpression with terminus v, the top subexpression.Any rex move starting at v, including the identity map, is a valid light leaf for the top subexpression, and all light leaves for the top subexpression are rex moves.Let us note that any other subexpression of v has terminus v ′ < v, so its light leaf lies in the ideal I <v .
If x = vz is a concatenation of two smaller sequences, we can restrict a subsequence e ⊂ x to a subsequence f ⊂ v. Suppose that v is a reduced expression for some v ∈ S n , and e restricts to the top subsequence f ⊂ v. Then we refer to LL e as a light tail, and since it is determined by the restriction of e to z, we may use the notation LT e\f .
Finally, suppose that x = yz is a concatenation of two smaller sequences, and e ∈ E(x, w) restricts to f ∈ E(y, v).Then LL f is a map from y to v for some reduced expression for v.Meanwhile, there is a light tail vz → w determined by the subexpression e \ f of z.The inductive construction of light leaves states that LL e = LT e\f •(LL f ⊗id z ).
(5.5) Schematically, we draw Because light tails will be important in some proofs below, let us introduce some terminology.
Definition 5.7.Let w ∈ S n .Relative to w, we call a sequence of simple reflections z = (s z1 , • • • , s zm ) a tail expression, and a sequence e ⊂ z of 0s and 1s a tail subexpression.A tail (sub)expression is the same thing as a (sub)expression, but we interpret its Bruhat stroll differently, and we decorate the subsequence with U s and Ds accordingly.There is a unique v ∈ S n such that vs e1 z1 s e2 z2 • • • s em z k = w.The tail Bruhat stroll associated to w and e ⊂ z is the Bruhat stroll which starts at v rather than 1, and wends its way to w.We call v the start of the tail Bruhat stroll.
Fix v, w ∈ S n and a reduced expression v for v.For a tail expression z, there is a bijection between ordinary subexpressions e ⊂ vz whose restriction to v is the top subsequence f ⊂ v, and tail subexpressions (e \ f ) ⊂ z with start v and terminus w.This bijection is natural over the choice of reduced expression for v, in the obvious sense.Henceforth (and unlike the previous paragraph), we always use (e \ f ) ⊂ z as notation for a tail subexpression, even though e and f themselves need not have been chosen.Whenever we choose another sequence y and consider the concatenation yz, and whenever f ∈ E(y, v), we will then set e to be the subexpression of yz whose restriction to y is f and whose restriction to z is e \ f .

Reminders: double leaves
For two sequences x and x ′ , subexpressions e ⊂ x and f ⊂ x ′ are called coterminal if they have the same terminus w, and we refer to (e, f , w) as a coterminal triple subordinate to (x, x ′ ).Sometimes we omit w from the triple, writing only (e, f ).To each coterminal triple we associate a double leaf morphism in Hom(x, x ′ ), where N is some rex move from the target of LL e to the source of ΓΓ f , both being reduced expressions for w.By taking one double leaf for each coterminal triple, one obtains a basis for Hom(x, x ′ ) as a left (or right) R n -module.Double leaves are always diagrams without polynomials, so they are killed by z.
Light leaves and double leaves are not determined only by the subexpressions e and f .There are many choices of rex moves in the construction of each light leaf, as well as the choice of rex move N in (5.7).In particular, the composition N • LL e of a rex move with a light leaf is, itself, another valid choice of a light leaf associated to e.In this way, we could remove the rex move N from (5.7), absorbing it into the light leaf LL e (or into the upside-down light leaf ΓΓ f ).
When one speaks about the double leaves basis, one must choose one amongst the many possible double leaves for each coterminal triple to be a basis element.There are many different double leaves bases.When we speak of light leaves or double leaves, we typically refer to the set of all possible maps produced by the algorithm, with the flexibility of using arbitrary rex moves.When we speak of a distinguished light leaf or double leaf, we must have fixed one for each subexpression, and we refer to that one.In this way we can separate in our language between the rigid choices one must make to get a basis, and the flexible choices which are sufficient for a spanning set.
There is a filtration by the spans of certain double leaves with regards to a certain partial order, for which the image of a double leaf in the associated graded does not depend on the choice of rex moves!This result will be proven in the next section.Thankfully, d will preserve this (downfree) filtration, and one can easily compute its action on the associated graded.One can not expect much more from the combinatorics of subexpressions: since d acts nontrivially on rex moves, one should not expect a formula for the d action on a double leaf which is independent of the choice of rex move.

The lexicoBruhat order
In this section we develop some technology for working with light leaves and double leaves.This technology is not original to this paper: it is part of work in progress [Eli] by the first author, and much was known (but not written in the literature) previously to the experts9 .
To study double leaves it helps to first study light leaves.From the fact that distinguished double leaves form a basis for Hom(x, w), we deduce10 that distinguished in E(x, w) form a basis for Hom(x, w)/I <w .Until further notice, we will be studying this Hom space, modulo lower terms.In the following proposition we discuss not the span of particular light leaves, but the span of all light leaves which are constructible by the non-deterministic algorithm of [EW16b, Construction 6.1].In other words, in this proposition we are agnostic to the choice of rex moves.
Proposition 5.8.Let w be a reduced expression for some w ∈ S n .Let x = yz be the concatenation of a sequence y of length k and a sequence z of length d − k.Each subexpression of x restricts to a subexpression of y.For v ∈ S n , let X ≤v ⊂ E(x, w) denote the subset which restricts to E(y, v ′ ) for some v ′ ≤ v. Let H ≤v denote the subspace of Hom(x, w) spanned (over the left action of R n ) by all possible light leaves corresponding to subexpressions in X ≤v .
Let f : y → v be any morphism whose target is a reduced expression v for v ∈ S n .Then for all g ∈ Hom(vz, w).Schematically, we have where v ′ ≤ v.
Proof.We suppose the result is true for all elements less than v in the Bruhat order, and deduce the result for v.Then, by induction, the result will be true for any v ∈ S n (without the need to check any base case).Note that everything else (i.e.y, z, w, k) is unchanging in this induction.We know that f ∈ Hom(y, v) and g ∈ Hom(vz, w) are both in the left R n -span of double leaves.So, up to taking left linear combinations over R n , our morphism has the following form.The source w ′ of ΓΓ 2 is a reduced expression for some element w ′ ≤ w.Thus either ΓΓ 2 ∈ I <w , or ΓΓ 2 is a rex move, and can be absorbed into LL 2 .We assume henceforth that ΓΓ 2 is the identity map, and w ′ = w.
The source v ′ of ΓΓ 1 is a reduced expression for some element v ′ ≤ v.If v ′ < v, then we can replace g with LL 2 •(ΓΓ 1 ⊗ id z ) and f with LL 1 , and we have a diagram of the form (5.8) but for v ′ instead of v.By induction, this lives in H ≤v ′ modulo I <w .So we reduce to the case when v ′ = v, in which case ΓΓ 1 is a rex move, and can be absorbed into LL 1 .Thus we have reduced to the following case.
Now LL 2 is associated to some subexpression e ⊂ vz, which restricts to some subexpression f ⊂ v.If f is the top subexpression (the unique subexpression with terminus v) then (5.11) is itself a light leaf by (5.6).If f is any other expression, then LL f sends v to a reduced expression for an element v ′ < v. Now we again can refactor our diagram, letting f = LL f • LL 1 , and letting g denote LT e\f , and can use the inductive hypothesis for v ′ .
Our main application of Proposition 5.8 will be to a mistaken light leaf, or a light leaf with error.Suppose that x = yz is a concatenation, and e ⊂ x.In order to construct LL e as in (5.5), we have already constructed the light leaf LL f associated to the restriction of e to y, and the light tail LT e\f associated to the restriction of e to z.However, instead of gluing these together (along the reduced expression v) as in (5.5), we make an error and insert some morphism E ∈ I <v as in the following picture.(5.12) The result is a mistaken light leaf; if y has length k, we say the mistake happened in the k-th tier of the light leaf, and we denote the mistaken light leaf by LL oops,k e .Lemma 5.9.With the same setup as for (5.12), the mistaken light leaf lies in the subspace H <v of Hom(x, w)/I <w .
Proof.This is immediate from Proposition 5.8.This lemma implies that LL oops,k e is in the span of light leaves LL e ′ which, at the k-th step, factor through elements < v rather than through v.However, we have not yet proven that the Bruhat stroll of e ′ does not go much higher than e ever went.Below we will get more control on mistaken light leaves, asserting that e ′ is less than e in some partial order on subexpressions.To get this additional control, we can not merely apply Proposition 5.8, but must produce a more subtle version.The reader should think of Proposition 5.8 as a warm-up exercise; the same ideas factor into the proof of Theorem 5.13.Definition 5.10.Let x = (s x1 , . . ., s x d ) be an expression of length d, and let e and e ′ be two subexpressions of x.Let (1 = w 0 , w 1 , . . ., w d ) and (1 = w ′ 0 , . . ., w ′ d ) be their associated Bruhat strolls, see [EW16b,§2.4].If e = e ′ , then let 1 ≤ k ≤ d be the index such that w k = w ′ k and w j = w ′ j for all j > k; we call k the index of last difference.We say that e ≺ e ′ if w k < w ′ k in the Bruhat order.We write e e ′ if either e = e ′ or e ≺ e ′ .We call this the lexicoBruhat order on subexpressions.
Lemma 5.11.The lexicoBruhat order is a total order on the set of subexpressions of x.
Proof.Clearly this relation is transitive, and e e ′ e implies e = e ′ .Suppose that e = e ′ , and let k be the index of last difference.Then w k = w ′ k s x k , so w k and w ′ k are comparable in the Bruhat order.Hence e and e ′ are comparable in the lexicoBruhat order.Remark 5.12.Fix w, and consider a tail expression z.Clearly the lexicoBruhat order on subexpressions can be extended to a total order on tail subexpressions of z relative to w, in the obvious way.Theorem 5.13.Fix w ∈ S n with a rex w, and let z be a tail expression of length m relative to w.Let e \ f ⊂ z be a tail subexpression with start v; choose a rex v for v, and a light tail LT e\f : vz → w.Let f : y → v be any morphism to v which lives in I <v .Then where H ≺e\f is the left R n -span of light leaves LL e ′ : yz → w whose tail subexpressions e ′ \ f ′ ⊂ z satisfy (e ′ \ f ′ ) ≺ (e \ f ) in the lexicoBruhat order.
Pictorially, we have ∈ H ≺e\f modulo I <w . (5.14) Before giving the proof, here is the immediate corollary.Suppose that the result holds for z, and let us extend z by prepending a simple reflection s.Suppose we have (e ⊂ f ) ⊂ sz, with start v. Let e \ f ′ be its restriction to z, which has start x.The associated light tail has a first tier LT 1 associated to s, and the remaining light tail LT 2 associated to z. Write f = f 1 • f 2 where the source of f 1 is a reduced expression for some element v ′ < v.We are analyzing the composition (5.16) The RHS is supposed to schematically represent a linear combination (over the left action of R n ) of morphisms of two kinds.In the second diagram on the RHS of (5.16), the curvy red morphism is supposed to represent the "lower terms" of Proposition 5.8.Note that f 2 can be absorbed into the lower terms, and the result exactly has the form (5.14) but for z rather than sz.By induction, this morphism is in the span of H ≺e\f ′ , which is a subspace of H ≺e\f , modulo I <w .
In the first diagram in (5.16), v ′′ is some element < v, but since it is the target of a light leaf whose source is a rex for v ′ , we must also have v ′′ ≤ v ′ < v.Meanwhile, the composition of the two light tails LT 2 •(LT ′ 1 ⊗id z ) is a light tail whose associated tail subexpression agrees with e \ f on z, but disagrees on s, going to a lower term.Let us resolve LL •f 2 , noting that light leaves span the maps to v ′′ modulo lower terms. (5.17) The first diagram on the RHS of (5.17) is a light leaf and lives in H ≺e\f .The second diagram is in H ≺e\f by induction on v, since v ′′ < v.This proves (5.14) for sz, completing the inductive step.
Corollary 5.15.Any two light leaves associated to the same subexpression e ∈ E(x, w) are equal in the space Hom(x, w)/I <w , modulo the left R n -span of light leaves for smaller subexpressions in the lexicoBruhat order.
Proof.Let LL and LL ′ be the two light leaves associated to e.The only difference between LL and LL ′ is the choice of rex moves at various tiers in the light leaves algorithm.Let x have length m.We can write LL − LL ′ as a telescoping sum where LL 0 = LL, LL ′ = LL m , each LL k is a light leaf for e, and LL k−1 differs from LL k in the choice of rex move made at the k-th tier of the algorithm.For example, LL m−1 uses the first m − 1 tiers of LL ′ , but uses the last tier from LL.
The difference between two rex moves (between two reduced expressions for some v ∈ S n ) consists of lower terms (i.e.lives in the ideal I <v ).Thus each difference LL k − LL k−1 is a mistaken light leaf, where the mistake happened in the k-th tier.By Corollary 5.13, LL k − LL k−1 is in the span of light leaves LL e ′ with e ′ ≺ e.
Remark 5.16.This corollary gives an alternate route to proving that distinguished light leaves span the space Hom(x, w)/I <w .Once one proves that light leaves span, this corollary proves that a single distinguished light leaf LL e will have the same span as all light leaves associated to e, modulo I <w and modulo light leaves for e ′ ≺ e.By induction on the lexicoBruhat order, one deduces that distinguished light leaves have the same span as all light leaves modulo I <w .
Let us quickly remark on the difference between the left R n -span and the right R n -span.Modulo lower terms, there is none!Thus all the results above also apply to the right R n -span.
Lemma 5.17.The left R n -span of any set of morphisms in Hom(x, w) agrees with the right R n -span modulo I <w .
Proof.Any polynomial p can be forced through the reduced expression for w at the top of the diagram, using (5.2).The result is w −1 (p) on the right hand side, plus error terms where strands are broken.These error terms are all in I <w .Now we bootstrap these results about light leaves to results about double leaves.Definition 5.18.Let x and y be expressions.Suppose that (e, f , w) and (e ′ , f ′ , w ′ ) are two coterminal triples subordinate to (x, y).We say that (e, f , w) (e ′ , f ′ , w ′ ) if e e ′ and f f ′ .We call this the lexicoBruhat order on triples.Suppose w < w ′ .If x has length m then the index of last difference between e and e ′ is m, and w < w ′ , so e ≺ e ′ .Similarly, f ≺ f ′ .Thus (e, f , w) ≺ (e ′ , f ′ , w ′ ) whenever w < w ′ .If particular, the span of all double leaves associated to triples less than (e, f , w) in the lexicoBruhat order will have I <w as a subspace.
The definition of a mistaken double leaf DL oops e,f is similar to that of a mistaken light leaf.At one tier in either LL e or ΓΓ f , one inserts an error term which goes lower in the Bruhat order than it should.
and (f , e ′ , w) ≺ (f , e, w) when e ′ ≺ e.Thus every term in the sum (5.20), when composed with ΓΓ f , is in the span of lower double leaves in the lexicoBruhat order on triples.
Corollary 5.20.Any two double leaves associated to the same triple (e, f , w) are equal in Hom(x, y) modulo the span of double leaves for smaller triples in the lexicoBruhat order.
Proof.The proof is the same as Corollary 5.15.
Remark 5.21.Once one proves that double leaves span, one deduces from Corollary 5.20 that distinguished double leaves have the same span.This gives an alternative proof of some of the results from [EW16b, Chapter 7].
To obtain an analog of Theorem 5.19 for the right R n -action, one can either use Lemma 5.17 and modify the proof of the theorem accordingly, or one can use the following lemma.
Proof.Consider any coterminal triple (e ′ , f ′ , w ′ ) ≺ (e, f , w).Clearly w ′ ≤ w.Using (5.2) to push a polynomial p across the reduced expression for w ′ in the middle of the double leaf, we get DL e ′ ,f ′ •p − w ′ (p) • DL e ′ ,f ′ ∈ I <w ′ .
(5.22)However, I <w ′ ⊂ I <w is already in the (right or left) span of lower double leaves.

Double leaves and the raising operator
The key result of this section will be that d(DL e,f ) is equal to p DL (e, f ) • DL e,f for some polynomial p(e, f ), plus a linear combination of mistaken double leaves.By Theorem 5.19, this means that d preserves the span of double leaves (or distinguished double leaves) associated to triples (e, f , w). (5.23c) To compute z(s λ ), recall that one can define the Schur polynomial as (6.9) By the above Corollary, z commutes with ∂ w0 .Applying z to the monomial inside the Demazure operator, we obtain If removing a box from the i-th row of λ yields a partition µ, then λ i − i is the content of that box.Applying ∂ w0 to the sum in (6.10), such terms yield λ=µ+ (c( ) + n)s µ which is the desired answer.We need only prove that all the other terms in (6.10) are killed by ∂ w0 .
If removing a box from the i-th row of λ is not a partition, it is because λ i = λ i+1 , in which case λ i − 1 + n − i = λ i+1 + n − (i + 1).Thus the coefficients of x i and x i+1 in the i-th monomial from the sum in (6.10) will be equal.Thus this monomial is invariant under s i , and killed by ∂ i and hence by ∂ w0 .Lemma 6.5.Divided powers d (k) and z (k) are defined integrally on R Sn n for all k ≥ 0.
Proof.These divided powers are defined on R n , and R Sn n is an sl 2 -invariant subring.

Actions on symmetric functions
It was observed in [EQ16b, end of §2.2] that the formulas for d acting on symmetric polynomials were independent of the number of variables n, and could be extended to a derivation on the ring Λ of symmetric functions (in infinitely many variables).However, it is clear from (6.6b) that the formulas for z acting on symmetric polynomials are not independent of n.If we let the number of variables go to infinity, where should z send e 1 = x 1 + x 2 + . .., given that z(x i ) = 1 for all i?The appropriate answer is to treat n as a formal variable; this was suggested to the authors by M. Khovanov.Proof.The existence of Ψ n for all n is immediate from the formulas for the actions of d and z.We can use this to efficiently check (6.12) using (6.6).For an example of this argument, consider z(h k ) ∈ Λ[y].This will be some [y]-linear combination of polynomials in the variables {h i }.Because Ψ n is equivariant, the coefficient of h k−1 is some polynomial which evaluates to n − 1 + k after specializing y → n for all n ≥ k.Thus this coefficient is precisely y − 1 + k.Similarly, the coefficient of h 1 h k−2 is some polynomial which evaluates to zero after specializing y → n for all n ≥ k.Thus this coefficient is zero.
Remark 6.8.For any n ∈ there is a quotient of Λ[y] which sends y → −n.It is an sl 2 -algebra over .
When n > 1, the ideal in this quotient generated by e k for k > n is sl 2 -invariant, and the further quotient by this ideal gives the map Ψ n .When n ≤ 0, there are no nontrivial sl 2 -invariant ideals in the quotient.
Sadly, divided powers of z are not defined integrally on Λ[y].For example, z (2) (e 5 ) = (y−3)(y−4) 2 e 3 , and y−3 2 := (y−3)(y−4) 2 is not in [y].The following lemma will imply that divided powers of z are defined integrally whenever y is specialized to an integer, even a negative integer.Lemma 6.9.On Λ[y], the divided powers d (m) are defined integrally, but z (m) are not defined integrally.However, if we base change over [y] to a larger base ring containing y+k m for any k ∈ and m ≥ 0, then z (m) is defined after base change.
Proof.We need only check that divided powers are well-defined on the generators.The formula for d does not involve the variable y, so the fact that d (m) (e k ) is defined integrally can be checked after applying Ψ n for sufficiently large n.Using (6.11b) it is easy to verify that z (m) (e k ) = y + 1 − k m e k−m .(6.14) Remark 6.10.Note that the divided powers z (m) are always defined integrally on power sums p k , without the need for binomial coefficients.However, power sums generate a different integral form for Λ[y] than do the elementary or complete symmetric functions.

Rank-one modules and their cores
In §2.5 we examined (R n , sl 2 )-modules which were free of rank 1 over R n .Now we do the same for (R Sn n , sl 2 )-modules and Λ[y]-modules.As before, the rank-one free modules are parametrized by degree two elements in the corresponding ring.
For each a ∈ define a rank-one module over R Sn n called R Sn n ae 1 , with generator 1 a living in degree an.Equip it with an sl 2 action by the formulas d(1 a ) = ae 1 1 a , z(1 a ) = 0. (6.15) For each a ∈ define a rank-one module over Λ[y] called Λ[y] ae 1 , with generator 1 a living in degree ay.Equip it with an sl 2 action by the same formulas (6.15).Note that this sl 2 -representation has weights not in but in [y].After specializing y to an integer, this yields an sl 2 -representation with weights in .We need to check that [z, d] = h on each of the generating morphisms of U, which is immediate from the formulas of [EQ16a, Definition 5.9].
We need to check that divided powers are defined integrally on the generating morphisms of U.For the generators inside a nilHecke algebra, this was already done in Theorem 4.4.The cups and caps are killed by z.All that remains to check is that d (k) is well-defined on the cups and caps.A related question was pursued in [EQ16a, Lemma A.3 and preceding], which proves that d p = 0 in characteristic p.The proof was to argue that all the coefficients appearing in d k were multinomial coefficients times k!, and this is the same proof 12 needed to show that d (k) is defined integrally.

Remarks on a downfree filtration
Let E denote the upward strand, an object in U, and F the downward strand.Throughout this section we fix λ ∈ and let Λ denote the specialization of Λ[y] at y = λ.
In [Lau10, §8], Lauda proves that Hom U (½ λ E n , ½ λ E n ) is isomorphic to Λ ⊗ NH n , where bubbles appear on the left of crossing diagrams.We can view crossing diagrams as a basis of this Hom space over Λ ⊗ R n , where R n acts on the bottom.Then this is a downfree filtration over the base ring Λ ⊗ R n , with the same downfree character computed in (4.17).This is because applying d to a crossing diagram will not create any bubbles.
Using adjunction, every morphism in the space Hom U (E n F n ½ λ , ½ λ ) is obtained by taking a morphism in Hom U (½ λ E n , ½ λ E n ) and placing caps on top.Thus crossing diagrams will form a basis for Hom U (E n F n , ½) under the action of Λ on the left and R n acting on the inwardly-pointing boundary strands.This basis is again in bijection with S n .However, the differential of a cap does introduce bubbles.We expect that this produces a downfree filtration over Λ ⊗ R n , but the formula for the downfree character is currently unknown, and will involve e 1 ∈ Λ.
For example, Hom U (EF ½ λ , ½ λ ) has a basis with one diagram, given by the cap.By [EQ16a, Definition 5.9], d sends this cup to itself with a dot minus itself with a degree two bubble.Thus this Hom space, as a module over the sl 2 -algebra R 1 ⊗ Λ, is isomorphic to (R 1 ⊗ Λ) x 1 − e 1 .
Similar arguments to those used in [Lau10, §8] will produce a basis for any morphism space in U.The basis will be a collection of reduced diagrams for oriented planar matchings with 2n boundary points (n oriented in, and n oriented out).It is a basis over the left action of Λ and the action of R n by placing dots on the inwardly-pointing boundary strands.The basis is in bijection with S n , though this bijection does not preserve the number of crossings in a diagram.We expect that this is a downfree basis, with a partial order coming from crossing removal; in this case, the partial order does not coincide with the usual Bruhat order on S n .The downfree character is currently unknown.It can be hard to determine from [KLMS12] what precisely the generators and relations of U are, over and above the presentation of U, as it is not stated explicitly there.We give the answer in [EQ16b, Proposition 5.2].There are two new generators for each thickness a: the splitter E (a) → E ⊗a and the merger E ⊗a → E (a) .(7.1) 12 Admittedly, the proof in [EQ16a, Lemma A.3] is rather hand-wavey, but the result is still a relatively easy exercise.

Theorem 1. 25 .
Let n = 1, so that R n = [x].Let M = [x] ax ⊕ [x] bx be an ( [x], sl 2 )-module with a downfree filtration, where [x] bx is the submodule.If the downfree filtration does not split then b = a+ 2. Unless a = 0 and b = 2, we have Core(M ) ∼ = Core( [x] ax ) ⊕ Core( [x] bx ).Proof.(Sketch) Let 1 a denote the generator of [x] ax , living in degree a. Then z(1 a ) = 0, d(1 a ) = ax1 a + m (1.22) for some m ∈ [x] bx .For the sl 2 relations to hold we need z(m) = 0.But the kernel of z inside [x] b is just the span of the generator 1 b .Thus m is a scalar multiple of 1 b , and for degree reasons b = a + 2.
The decomposition X ∼ = Y ⊕ Z implies that Hom(X, −) ∼ = Hom(Y, −) ⊕ Hom(Z, −), or more precisely Hom(X, −) ∼ = Hom(Y, −) • p Y ⊕ Hom(Z, −) • p Z , (3.11) where p Y and p Z are the projection maps.Consider Hom(X, Y ) and Hom(X, Z).Both are supported in non-negative degrees, and are spanned in degree zero by their respective projection maps p Y and p Z .Thus for degree reasons, z(p Y ) = 0 and z(p Z ) = 0, and if either Hom space has a nonzero core, it must be a trivial module spanned by the projection map.The first miracle is that d(p Y ) = 0. (3.12)As a consequence, Hom(Y, −) • p Y ⊂ Hom(X, −) is preserved by the sl 2 action, making the decomposition (3.11) a filtration with respect to sl 2 .In particular, this submodule Hom(Y, −) • p Y is generated by the core of Hom(X, Y ).Meanwhile, d(p Z ) = 0, and Core(Hom(X, Z)) = 0.The second miracle is that d(p Z ) ∈ Hom(Y, Z) • p Y .(3.13)We have already observed by Hom(Y, Z) • p Y is preserved by sl 2 inside Hom(X, Z), and (3.13) implies that the image of p Z spans the core of the quotient module Hom(X, Z)/(Hom(Y, Z) • p Y ).

4
sl 2 action on the KLR algebra 4.1 Definition of the action Definition 4.1.Let U + (g) denote the Khovanov-Lauda-Rouquier category associated to an oriented simply laced root datum.A presentation by generators and relations can be found in [KQ15, §4.1].

Remark 4. 2 .
The original definition of Khovanov-Lauda[KL09] used Q ij = x i + x j , while the signed variant was introduced in [KL11, page 15].It was correctly predicted on [KL11, page 17] that this signed variant matches Lusztig's geometric categorification of the quantum group.As shown explicitly by Lauda in [Lau20, Proposition 3.4], different choices of Q ij yield isomorphic categories when the Dynkin diagram is simply connected.This isomorphism will rescale the polynomials x i by various signs.Evidently one could transfer our sl 2 action along Lauda's isomorphisms, but they would no longer satisfy (4.1).Definition 4.3.Equip U + (g) with an sl 2 action as follows.The derivation d is the derivation ∂ 1 defined in [KQ15, Definition 4.13].The derivation z sends a dot to 1, and kills all crossings.In formulas, where red and blue are adjacent colors and red and green are distant, we have That d preserves the relations was checked 8 in [KQ15, §4.1].Let us check that z preserves the relations.Any diagram without dots is sent to zero by z.Any relation which is a linear combination of diagrams without dots is therefore preserved by z.The remaining relations are these.z will kill the identity map, and send the other diagrams to the crossing, is preserved.The vertical flip is also a relation, and preserved for the same reason.Similarly, z will preserve the relation relation (e.g. when colors are distant, or the vertical flip) are preserved by z for the same reason.The final relation with dots involves two adjacent colors.
m v,w := #{c | a < c < b and w(b) < w(a) < w(c)}.(4.19) (a, b) crossing.Thus the coefficient involved in the removal of the (a, b) crossing is • +1 from (4.14), • +2 if some (c, b) crossing occurs below the (a, b) crossing, • −2 if some (c, a) crossing occurs below the (a, b) crossing.So let c be any other strand.If some (c, a) crossing occurs below the (a, b) crossing then clearly c < a < b.Since both (c, a) and (a, b) crossings appear, so must (c, b), and since (c, a) happens before (a, b), they come in lexicographic order.So (c, b) also occurs below (a, b).Then the overall contribution to the coefficient is −2 + 2 = 0. Thus no contribution of −2 can occur without being canceled by a contribution of +2.Conversely, if c < a < b and a (c, b) crossing occurs before an (a, b) crossing, then the crossings appear in lexicographic order, meaning that a (c, a) crossing must have come first.If some (c, b) crossing occurs below the (a, b) crossing, but a (c, a) crossing does not also occur below the (a, b) crossing, then we must have a < c < b.This will contribute +2 to the coefficient, and m v,w is exactly counting such contributions.

5
sl 2 action on the Hecke category 5.1 Definition of the action Definition 5.1.Let H = H(S n ) denote the diagrammatic Hecke category associated to the action of S n on R n = [x 1 , . . ., x n ].A presentation by generators and relations can be found in [EK10].
z has length 1, we think of LT e\f as being a single tier of the light leaf algorithm.When z has length d, the light tail is built inductively from the last d tiers.This is discussed in[EW16b, Remark 6.4].

Corollary 5. 14 .
Fix x = yz, where y has length k.Let e ∈ E(x, w), and let f be the restriction of e to y.Let LL oops,k e be any mistaken light leaf where the mistake happens in the k-th tier of the light leaf.Then LL oops,k e ∈ H ≺e\f modulo I <w .(5.15)Proof of Theorem 5.13.We prove this by induction on the length m of z, and within each m, by induction on v.If m = 0 then f ∈ I <w and the result is trivial.If m = 1 then the result actually follows easily Proposition 5.8, but we will use essentially the same argument below for the inductive step.

.
Now apply Proposition 5.8 to the subdiagram which is LT 1
. B. E. was partially supported by NSF CAREER grant DMS-1553032 and NSF FRG grant DMS-1800498.This paper was completed while B. E. was visiting the Institute of Advanced Study, where he was supported by NSF grant DMS-1926686.Y.Q. was partially supported by the NSF grant DMS-1947532.The authors would like to thank Mikhail Khovanov for his interest, and his suggestion in §6.2.
Theorem 5.19.Any mistaken double leaf DL oops e,f is in the left R n span of double leaves associated to triples (e ′ , f ′ , w ′ ) which are smaller than (e, f , w) in the lexicoBruhat order.Proof.The error in DL oops e,f is made either in LL e or in ΓΓ f , and the arguments are the same either way, so let us assume the error is made in LL e .Thus Since I <w is a two-sided ideal, ΓΓ f • I <w ⊂ I <w .Also, I <w lies in the span of smaller double leaves (as discussed a few paragraphs ago).Meanwhile,