JournalsjemsVol. 15, No. 3pp. 775–803

Invariants for the modular cyclic group of prime order via classical invariant theory

  • David L. Wehlau

    Royal Military College of Canada, Kingston, Canada
Invariants for the modular cyclic group of prime order via classical invariant theory cover
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Abstract

Let F\mathbb F be any field of characteristic pp. It is well-known that there are exactly pp inequivalent indecomposable representations V1,V2,,VpV_1,V_2,\dots,V_p of CpC_p defined over F\mathbb F. Thus if VV is any finite dimensional CpC_p-representation there are non-negative integers 0n1,n2,,nkp10\leq n_1,n_2,\dots, n_k \leq p-1 such that Vi=1kVni+1V \cong \oplus_{i=1}^k V_{n_i+1}. It is also well-known there is a unique (up to equivalence) d+1d+1 dimensional irreducible complex representation of SL2(C)_2(\mathbb C) given by its action on the space RdR_d of dd forms. Here we prove a conjecture, made by R.~J.~Shank, which reduces the computation of the ring of CpC_p-invariants F[i=1kVni+1]Cp\mathbb F[ \oplus_{i=1}^k V_{n_i+1}]^{C_p} to the computation of the classical ring of invariants (or covariants) C[R1(i=1kRni)]SL2(C)\mathbb C[R_1 \oplus (\oplus_{i=1}^k R_{n_i})]^{\mathrm {SL}_2(\mathbb C)}. This shows that the problem of computing modular CpC_p invariants is equivalent to the problem of computing classical SL2(C)_2(\mathbb C) invariants. This allows us to compute for the first time the ring of invariants for many representations of CpC_p. In particular, we easily obtain from this generators for the rings of vector invariants F[mV2]Cp\mathbb F[m\,V_2]^{C_p}, F[mV3]Cp\mathbb F[m\,V_3]^{C_p} and F[mV4]Cp\mathbb F[m\,V_4]^{C_p}for all mNm \in \mathbb N. This is the first computation of the latter two families of rings of invariants.

Cite this article

David L. Wehlau, Invariants for the modular cyclic group of prime order via classical invariant theory. J. Eur. Math. Soc. 15 (2013), no. 3, pp. 775–803

DOI 10.4171/JEMS/376