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arXiv · 2412.06524

Equivariant Ehrhart Theory of Hypersimplices

Oliver Clarke·Max Kölbl
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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.

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Oliver Clarke, Max Kölbl. 2025-03-03. Equivariant Ehrhart Theory of Hypersimplices. https://doi.org/10.1017/fms.2025.10124

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