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Order polytopes of graded posets are gamma-effective

Alessio D'Alì, Akihiro Higashitani

TL;DR

The paper extends Brändén's γ-positivity for order polytopes to an equivariant setting by introducing order polytopes of sign-graded posets and developing an equivariant Ehrhart framework. A key innovation is a saturation-based formula that expresses the equivariant h^*-series h^*_P,ε^G(t) as an induced sum over stabilizers of saturations of (P,ε). The main theoretical breakthrough is proving γ-effectiveness for order polytopes of graded posets under any finite group action, connecting equivariant γ-polynomials to cubes and their known representation-theoretic γ-coefficients. The work provides concrete computational tools, including an explicit example with a D_4-action, and broadens the applicability of γ-positivity in discrete geometry and combinatorics, with potential implications for symmetry-aware lattice polytopes and their Ehrhart theory.

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 $γ$-nonnegative by work of Brändén, who introduced the concept of sign-graded poset in the process. In the present paper we are interested in proving an equivariant version of Brändén'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 $γ$-effective, i.e., that the $γ$-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 (à la Brändén) of the given sign-graded poset.

Order polytopes of graded posets are gamma-effective

TL;DR

The paper extends Brändén's γ-positivity for order polytopes to an equivariant setting by introducing order polytopes of sign-graded posets and developing an equivariant Ehrhart framework. A key innovation is a saturation-based formula that expresses the equivariant h^*-series h^*_P,ε^G(t) as an induced sum over stabilizers of saturations of (P,ε). The main theoretical breakthrough is proving γ-effectiveness for order polytopes of graded posets under any finite group action, connecting equivariant γ-polynomials to cubes and their known representation-theoretic γ-coefficients. The work provides concrete computational tools, including an explicit example with a D_4-action, and broadens the applicability of γ-positivity in discrete geometry and combinatorics, with potential implications for symmetry-aware lattice polytopes and their Ehrhart theory.

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 -nonnegative by work of Brändén, who introduced the concept of sign-graded poset in the process. In the present paper we are interested in proving an equivariant version of Brändén'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 -effective, i.e., that the -polynomial associated with the equivariant -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 (à la Brändén) of the given sign-graded poset.
Paper Structure (16 sections, 17 theorems, 78 equations, 5 figures, 2 tables)

This paper contains 16 sections, 17 theorems, 78 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Ehrhart theory, gamma-nonnegativity, and their equivariant counterparts
  3. Main results
  4. Structure of this paper
  5. Sign-graded posets and their automorphisms
  6. Notation and first notions
  7. Quotient posets
  8. Saturations and their interaction with quotient posets
  9. Order polytopes of sign-graded posets and their equivariant Ehrhart series
  10. Order polytopes of sign-graded posets
  11. Representation theory of finite groups
  12. Equivariant Ehrhart theory for order polytopes
  13. Equivariant gamma-polynomials of graded posets
  14. Equivariant Ehrhart series of cubes and their gamma-effectiveness
  15. Gamma-effectiveness for arbitrary order polytopes
  16. ...and 1 more sections

Key Result

Theorem 1.3

If $P$ is a graded poset, then the $h^*$-polynomial of $\mathscr{O}(P)$ is (palindromic and) $\gamma$-nonnegative.

Figures (5)

  • Figure 1: The poset $P$.
  • Figure 2: The action by $\sigma$.
  • Figure 3: The action by $\tau$.
  • Figure 4: The saturations of $(P,\mathbf{1}) = (P,\varepsilon_{\mathrm{par}})$.
  • Figure 5: The $G$-orbit of the saturation $(Q_2, \varepsilon_{\mathrm{par}})$.

Theorems & Definitions (41)

  • Definition 1.1
  • Theorem 1.3: branden04
  • Theorem A: see Theorem \ref{['thm:main theorem on h^*']}
  • Theorem B: see Theorem \ref{['thm:gamma-effective']}
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • ...and 31 more