Universal Hilbert series coefficients of the superspace coinvariant ring

Abstract

The coefficients that determine the Hilbert series of the superspace coinvariant ring are indexed by hook-shaped partitions. We give a manifestly positive combinatorial interpretation of these coefficients, together with several generating functions for them. Specializing this Hilbert series at $u=-q^2$, we show that its coefficients are differences of binomial coefficients. Consequently, this proves a conjecture of Sagan--Swanson (2024) that these coefficients are palindromic up to sign. More generally, for every $m\ge 1$ we obtain closed-form expressions for the $u=-q^m$ specialization.

22 pages, 1 figure. Comments are welcome