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

David L. Wehlau

Journal of the European Mathematical Society, Tome 15 (2013) no. 3, pp. 775-803.

Résumé

Let F be any field of characteristic p. It is well-known that there are exactly p inequivalent indecomposable representations V1​,V2​,...,Vp​ of Cp​ defined over F. Thus if V is any finite dimensional Cp​-representation there are non-negative integers 0≤n1​,n2​,...,nk​≤p−1 such that V≅⊕i=1k​Vni​+1​. It is also well-known there is a unique (up to equivalence) d+1 dimensional irreducible complex representation of SL2​(C) given by its action on the space Rd​ of d forms. Here we prove a conjecture, made by R. J. Shank, which reduces the computation of the ring of Cp​-invariants F[⊕i=1k​Vni​+1​]Cp​ to the computation of the classical ring of invariants (or covariants) C[R1​⊕(⊕i=1k​Rni​​)]SL2​(C). This shows that the problem of computing modular Cp​ invariants is equivalent to the problem of computing classical SL2​(C) invariants. This allows us to compute for the first time the ring of invariants for many representations of Cp​. In particular, we easily obtain from this generators for the rings of vector invariants F[mV2​]Cp​, F[mV3​]Cp​ and F[mV4​]Cp​for all m∈N. This is the first computation of the latter two families of rings of invariants.

Détail

DOI : 10.4171/jems/376

Classification : 13-XX, 20-XX, 00-XX
Keywords: Modular invariant theory, cyclic group, classical invariant theory, Roberts' isomorphism