Order polytopes of graded posets are gamma-effective

TL;DR

This paper proves that order polytopes of graded posets are gamma-effective, meaning their equivariant h*-polynomials have coefficients that are actual characters, extending known gamma-nonnegativity results using equivariant Ehrhart theory.

Contribution

It introduces a theory of order polytopes of sign-graded posets and establishes their gamma-effectiveness through an equivariant Ehrhart theoretic approach.

Findings

Order polytopes of graded posets are gamma-effective.

Develops a formula for the equivariant Ehrhart series numerator.

Extends gamma-nonnegativity to an equivariant setting.

Abstract

Order polytopes of posets have been a very rich topic at the crossroads between combinatorics and discrete geometry since their definition by Stanley in 1986. Among other notable results, order polytopes of graded posets are known to be $\gamma$-nonnegative by work of Br\"and\'en, who introduced the concept of sign-graded poset in the process. In the present paper we are interested in proving an equivariant version of Br\"and\'en's result, using the tools of equivariant Ehrhart theory (introduced by Stapledon in 2011). Namely, we prove that order polytopes of graded posets are always $\gamma$-effective, i.e., that the $\gamma$-polynomial associated with the equivariant $h^*$-polynomial of the order polytope of any graded poset has coefficients consisting of actual characters. To reach this goal, we develop a theory of order polytopes of sign-graded posets, and find a formula to express the numerator of the equivariant Ehrhart series of such an object in terms of the saturations (\`a la Br\"and\'en) of the given sign-graded poset.