Deletion formulas for equivariant Kazhdan-Lusztig polynomials of matroids

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 equ...

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Bibliographic Details
Published in:arXiv.org 2024-06
Main Authors: Ferroni, Luis, Matherne, Jacob P, Vecchi, Lorenzo
Format: Article
Language:English
Subjects:
Deletion
Polynomials
Online Access:Get full text
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Summary: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 \(\gamma\)-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 Schr\"oter.
ISSN:2331-8422