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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2602.10646 |
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| _version_ | 1866915791202418688 |
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| author | Xie, Matthew H. Y. Zhang, Philip B. Zhong, Michael X. X. |
| author_facet | Xie, Matthew H. Y. Zhang, Philip B. Zhong, Michael X. X. |
| contents | The thagomizer matroid, realized as the graphic matroid of the complete tripartite graph $K_{1,1,n}$, has full automorphism group isomorphic to the hyperoctahedral group whenever $n \ge 2$. In the equivariant setting for this action, we compute both the Kazhdan--Lusztig polynomial and the inverse Kazhdan--Lusztig polynomial in the sense of Proudfoot's Kazhdan--Lusztig--Stanley theory, and we show that each coefficient is an honest representation with a multiplicity-free irreducible decomposition. Our main idea is to exploit the palindromicity of the equivariant $Z$-polynomial, reducing the computation to the already established symmetric-group equivariant Kazhdan--Lusztig theory for the graphic matroids of cycle graphs, and then to apply Proudfoot's equivariant Kazhdan--Lusztig--Stanley inversion identity to obtain the inverse polynomial. Passing to dimensions recovers the previously known nonequivariant thagomizer polynomials, while the coefficient formulas admit a natural expression in terms of the wreath product Frobenius characteristic for the hyperoctahedral group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_10646 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Equivariant Kazhdan--Lusztig Polynomials of Thagomizer Matroids with a Hyperoctahedral Group Action Xie, Matthew H. Y. Zhang, Philip B. Zhong, Michael X. X. Combinatorics The thagomizer matroid, realized as the graphic matroid of the complete tripartite graph $K_{1,1,n}$, has full automorphism group isomorphic to the hyperoctahedral group whenever $n \ge 2$. In the equivariant setting for this action, we compute both the Kazhdan--Lusztig polynomial and the inverse Kazhdan--Lusztig polynomial in the sense of Proudfoot's Kazhdan--Lusztig--Stanley theory, and we show that each coefficient is an honest representation with a multiplicity-free irreducible decomposition. Our main idea is to exploit the palindromicity of the equivariant $Z$-polynomial, reducing the computation to the already established symmetric-group equivariant Kazhdan--Lusztig theory for the graphic matroids of cycle graphs, and then to apply Proudfoot's equivariant Kazhdan--Lusztig--Stanley inversion identity to obtain the inverse polynomial. Passing to dimensions recovers the previously known nonequivariant thagomizer polynomials, while the coefficient formulas admit a natural expression in terms of the wreath product Frobenius characteristic for the hyperoctahedral group. |
| title | Equivariant Kazhdan--Lusztig Polynomials of Thagomizer Matroids with a Hyperoctahedral Group Action |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2602.10646 |