Equivariant Ehrhart theory of hypersimplices

Clarke, Oliver  ; Kölbl, Max 

Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e178

Résumé

We study the hypersimplex under the action of the symmetric group $S_n$ by coordinate permutation. We prove that its equivariant volume, given by the evaluation of its equivariant $H^*$-series 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.

Détail

DOI : 10.1017/fms.2025.10124

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