Presentations of categories of modules using the Cautis–Kamnitzer–Morrison principle

Abstract

We study commuting actions of a complex reductive Lie algebra, $\mathfrak{g}$, and an algebra $A$ on a finite dimensional vector space $V$. We outline how such commuting actions can be used to give a presentation of the category of $A$-modules isomorphic to $\mathfrak{g}$-weight spaces of $V$. This generalizes a result of Cautis–Kamnitzer–Morrison (2014) in the non-quantum setting. As examples, we apply this to classical Schur–Weyl duality and the Brauer Schur–Weyl dualities. In particular we obtain a diagrammatic presentation, in terms of generators and relations, of the category of permutation modules of the symmetric group, $S_d$. This is used to give a new description of the Kronecker product of the $S_d$-equivariant morphisms between permutation modules.