Equivariant Ehrhart Theory of Hypersimplices

TL;DR

This paper investigates the equivariant Ehrhart theory of hypersimplices, proving a conjecture about the $H^*$-polynomial's evaluation and providing formulas for its coefficients, with implications for related conjectures.

Contribution

It proves Stapledon's conjecture for hypersimplices and introduces a formula for the $H^*$-polynomial coefficients, revealing new counterexamples to another conjecture.

Findings

Evaluation of $H^*$-polynomial at 1 equals permutation character

Formula for coefficients of the $H^*$-polynomial

Counterexamples to Stapledon's conjecture involving trivial characters

Abstract

We study the hypersimplex under the action of the symmetric group $S_n$ by coordinate permutation. We prove that the evaluation of its equivariant $H^*$-polynomial at $1$ is the permutation character of decorated ordered set partitions under the natural action of $S_n$. This verifies a conjecture of Stapledon for the hypersimplex. To prove this result, we give a formula for the coefficients of the $H^*$-polynomial. Additionally, for the $(2,n)$-hypersimplex, we use this formula to show that trivial character need not appear as a direct summand of a coefficient of the $H^*$-polynomial, which gives a family of counterexamples to a different conjecture of Stapledon.