Equivariant inverse $Z$-polynomials of matroids
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| Format: | Preprint |
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2026
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| author | Gao, Alice L. L. Li, Yun Xie, Matthew H. Y. |
| author_facet | Gao, Alice L. L. Li, Yun Xie, Matthew H. Y. |
| contents | Motivated by the notion of the inverse $Z$-polynomial introduced by Ferroni, Matherne, Stevens, and Vecchi, we study the equivariant inverse $Z$-polynomial of a matroid equipped with a finite group. We prove that the coefficients of the equivariant inverse $Z$-polynomials are honest representations and that these polynomials are palindromic. Explicit formulas are obtained for uniform matroids equipped with the symmetric group. The corresponding formulas for $q$-niform matroids are derived using the Comparison Theorem for unipotent representations. For arbitrary equivariant paving matroids, explicit expressions are obtained by relating the polynomials of a matroid to those of its relaxation. We show that these polynomials are equivariantly unimodal and strongly inductively log-concave for both uniform and $q$-niform matroids. Motivated by the properties of equivariant $Z$-polynomials, we conjecture that the coefficients of the equivariant inverse $Z$-polynomials are equivariantly unimodal and strongly equivariantly log-concave. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_17314 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Equivariant inverse $Z$-polynomials of matroids Gao, Alice L. L. Li, Yun Xie, Matthew H. Y. Combinatorics 05B35, 05E05, 20C30 Motivated by the notion of the inverse $Z$-polynomial introduced by Ferroni, Matherne, Stevens, and Vecchi, we study the equivariant inverse $Z$-polynomial of a matroid equipped with a finite group. We prove that the coefficients of the equivariant inverse $Z$-polynomials are honest representations and that these polynomials are palindromic. Explicit formulas are obtained for uniform matroids equipped with the symmetric group. The corresponding formulas for $q$-niform matroids are derived using the Comparison Theorem for unipotent representations. For arbitrary equivariant paving matroids, explicit expressions are obtained by relating the polynomials of a matroid to those of its relaxation. We show that these polynomials are equivariantly unimodal and strongly inductively log-concave for both uniform and $q$-niform matroids. Motivated by the properties of equivariant $Z$-polynomials, we conjecture that the coefficients of the equivariant inverse $Z$-polynomials are equivariantly unimodal and strongly equivariantly log-concave. |
| title | Equivariant inverse $Z$-polynomials of matroids |
| topic | Combinatorics 05B35, 05E05, 20C30 |
| url | https://arxiv.org/abs/2601.17314 |