Equivariant Ehrhart Theory of Hypersimplices

2024/12/09 by Clarke, Oliver, Kölbl, Max
#05E10 #20C10 (secondary) #52B15 #52B20 (primary) 05E18 #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2412.06524

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.

Related

• The h^*-polynomials of type C hypersimplices
• A combinatorial formula for the Ehrhart h^*-vector of the hypersimplex
• Conjectures for Ehrhart h^*-vectors of Hypersimplices and Dilated Simplices
• Techniques in equivariant Ehrhart theory
• Ehrhart h^*-vectors of hypersimplices