Lewis–Reiner–Stanton Hilbert Series Conjecture for GL_n-invariants of truncated polynomial rings

Determine, for every finite field F_q, integers n ≥ 1 and m ≥ 1, whether the (q,t)-Hilbert series of the GL_n(F_q)-invariant subspace of the truncated polynomial ring Q = F_q[x_1, …, x_n]/(x_1^{q^m}, …, x_n^{q^m}) equals the Lewis–Reiner–Stanton polynomial C_{n,m}(t) = ∑_{k=0}^{min(n,m)} t^{(n−k)(q^m − q^k)} (m choose k)_{q,t}, where (m choose k)_{q,t} denotes the (q,t)-multinomial coefficient.

Background

The paper studies modular invariants of truncated polynomial rings Q = F_q[x_1, …, x_n]/(x_1^{q^m}, …, x_n^{q^m}) under the action of GL_n(F_q). Lewis–Reiner–Stanton proposed a conjectural formula for the graded Hilbert series of invariant subspaces, expressed via (q,t)-multinomial coefficients. This conjecture has been verified in several low-rank cases and for certain parabolic subgroups, but its full generality remains part of an ongoing program.

In this work, the authors confirm the conjecture for rank four under an explicit technical hypothesis, but the conjecture itself is stated in general form in the introduction as motivation for the paper. The extracted statement captures the precise formulation for the full general linear group case α = (n), including the explicit series formula.