Invariants of polynomials mod Frobenius powers

Abstract

Lewis, Reiner, and Stanton conjectured a Hilbert series for a space of invariants under an action of finite general linear groups using (q, t)-binomial coefficients. This work gives an analog in positive characteristic of theorems relating various Catalan numbers to the representation theory of rational Cherednik algebras. They consider a finite general linear group as a reflection group acting on the quotient of a polynomial ring by iterated powers of the irrelevant ideal under the Frobenius map. We prove a variant of their conjecture in the local case, when the group acting fixes a reflecting hyperplane, over fields of prime order. m [q m ] := (x q m 1 , . . . , x q m n ) , which we call the Frobenius irrelevant ideal. Their conjecture gives the Hilbert series for the GL n (F q )-invariants in F q [x 1 , . . . , x n ]/(x q m 1 , . . . , x q m n ) using (q, t)-binomial coefficients. We consider subgroups of reflections about a single hyperplane H in V . These groups are not cyclic in general, in contrast to groups over fields of characteristic 0. We take the case when q is a prime p; some of our ideas generalize to arbitrary q. We explicitly describe the space of G-invariants in S/m [p m ] for any subgroup G ⊂ GL n (F p ) fixing a hyperplane H in V pointwise. We give the Hilbert series in terms of the dimension of the transvection root space. As a special case, we describe the invariants under the pointwise stabilizer GL n (F p ) H in GL n (F p ) of any hyperplane H in V .