The depth of rings of invariants over finite fields

Abstract

We study the depth of S(V)^G, the ring of elements in the symmetric algebra of an n-dimensional vector space over a finite field which are invariant under the action of a subgroup G≤GL(V). The ring S(V)^G is a finite extension of the ring of invariants S(V)^GL(V), which is a polynomial ring on the Dickson invariants u_r=c_n,n−r ([2], [12]). We conjecture that the depth of S(V)^G is the largest r such that u₁,...,u_r is a regular sequence on S(V)^G, and show this to be true if depth S(V)^G is 1, 2, n−1 or n. We also give a proof, using Steenrod operations, that over a prime field , depth S(V)^G≥3 implies u₁, u₂, u₃ is a regular sequence on S(V)^G.