Equivariant Ehrhart Theory of Hypersimplices
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911373042122752 |
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| author | Clarke, Oliver Kölbl, Max |
| author_facet | Clarke, Oliver Kölbl, Max |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_06524 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Equivariant Ehrhart Theory of Hypersimplices Clarke, Oliver Kölbl, Max Combinatorics Representation Theory 52B20 (primary) 05E18, 52B15, 05E10, 20C10 (secondary) 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. |
| title | Equivariant Ehrhart Theory of Hypersimplices |
| topic | Combinatorics Representation Theory 52B20 (primary) 05E18, 52B15, 05E10, 20C10 (secondary) |
| url | https://arxiv.org/abs/2412.06524 |