Equivariant Ehrhart Theory of Hypersimplices

We study the hypersimplex under the action of the symmetric group $S_n$ by coordinate permutation. We prove that the evaluation of its equivariant $H^{\ast }$-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 $H^{\ast }$-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 $H^{\ast }$-polynomial, which gives a family of counterexamples to a different conjecture of Stapledon.