Order polytopes of graded posets are gamma-effective
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908360580792320 |
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| author | D'Alì, Alessio Higashitani, Akihiro |
| author_facet | D'Alì, Alessio Higashitani, Akihiro |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07623 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Order polytopes of graded posets are gamma-effective D'Alì, Alessio Higashitani, Akihiro Combinatorics Representation Theory Primary: 05E18, Secondary: 52B20, 06A07, 52B15 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. |
| title | Order polytopes of graded posets are gamma-effective |
| topic | Combinatorics Representation Theory Primary: 05E18, Secondary: 52B20, 06A07, 52B15 |
| url | https://arxiv.org/abs/2505.07623 |