Equivariant inverse Kazhdan--Lusztig polynomials of thagomizer matroids
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| Format: | Preprint |
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2025
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| _version_ | 1866912644667015168 |
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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 | In this paper, we focus on the equivariant inverse Kazhdan--Lusztig polynomials of thagomizer matroids, a natural family of graphic matroids associated with the complete tripartite graphs $K_{1,1,n}$. These polynomials were introduced by Proudfoot as an extension of the Kazhdan--Lusztig theory for matroids. We derive closed-form expressions for the $\mathfrak{S}_n$-equivariant inverse Kazhdan--Lusztig polynomials of thagomizer matroids and present them explicitly in terms of the irreducible representations of $\mathfrak{S}_n$. As an application, we also provide explicit formulas for the non-equivariant inverse Kazhdan--Lusztig polynomials, originally defined by Gao and Xie, and give an alternative proof using generating functions. Furthermore, we prove that the inverse Kazhdan--Lusztig polynomials of thagomizer matroids are log-concave. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_11322 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Equivariant inverse Kazhdan--Lusztig polynomials of thagomizer matroids Gao, Alice L. L. Li, Yun Xie, Matthew H. Y. Combinatorics 05B35, 05E05, 20C30 In this paper, we focus on the equivariant inverse Kazhdan--Lusztig polynomials of thagomizer matroids, a natural family of graphic matroids associated with the complete tripartite graphs $K_{1,1,n}$. These polynomials were introduced by Proudfoot as an extension of the Kazhdan--Lusztig theory for matroids. We derive closed-form expressions for the $\mathfrak{S}_n$-equivariant inverse Kazhdan--Lusztig polynomials of thagomizer matroids and present them explicitly in terms of the irreducible representations of $\mathfrak{S}_n$. As an application, we also provide explicit formulas for the non-equivariant inverse Kazhdan--Lusztig polynomials, originally defined by Gao and Xie, and give an alternative proof using generating functions. Furthermore, we prove that the inverse Kazhdan--Lusztig polynomials of thagomizer matroids are log-concave. |
| title | Equivariant inverse Kazhdan--Lusztig polynomials of thagomizer matroids |
| topic | Combinatorics 05B35, 05E05, 20C30 |
| url | https://arxiv.org/abs/2510.11322 |