Deletion formulas for equivariant Kazhdan-Lusztig polynomials of matroids
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Ferroni, Luis Matherne, Jacob P. Vecchi, Lorenzo |
| author_facet | Ferroni, Luis Matherne, Jacob P. Vecchi, Lorenzo |
| contents | We study equivariant Kazhdan--Lusztig (KL) and $Z$-polynomials of matroids. We formulate an equivariant generalization of a result by Braden and Vysogorets that relates the equivariant KL and $Z$-polynomials of a matroid with those of a single-element deletion. We also discuss the failure of equivariant $γ$-positivity for the $Z$-polynomial. As an application of our main result, we obtain a formula for the equivariant KL polynomial of the graphic matroid gotten by gluing two cycles. Furthermore, we compute the equivariant KL polynomials of all matroids of corank~$2$ via valuations. This provides an application of the machinery of Elias, Miyata, Proudfoot, and Vecchi to corank $2$ matroids, and it extends results of Ferroni and Schröter. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_19962 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Deletion formulas for equivariant Kazhdan-Lusztig polynomials of matroids Ferroni, Luis Matherne, Jacob P. Vecchi, Lorenzo Combinatorics Representation Theory We study equivariant Kazhdan--Lusztig (KL) and $Z$-polynomials of matroids. We formulate an equivariant generalization of a result by Braden and Vysogorets that relates the equivariant KL and $Z$-polynomials of a matroid with those of a single-element deletion. We also discuss the failure of equivariant $γ$-positivity for the $Z$-polynomial. As an application of our main result, we obtain a formula for the equivariant KL polynomial of the graphic matroid gotten by gluing two cycles. Furthermore, we compute the equivariant KL polynomials of all matroids of corank~$2$ via valuations. This provides an application of the machinery of Elias, Miyata, Proudfoot, and Vecchi to corank $2$ matroids, and it extends results of Ferroni and Schröter. |
| title | Deletion formulas for equivariant Kazhdan-Lusztig polynomials of matroids |
| topic | Combinatorics Representation Theory |
| url | https://arxiv.org/abs/2406.19962 |