Deletion formulas for equivariant Kazhdan-Lusztig polynomials of matroids

TL;DR

This paper develops deletion formulas for equivariant Kazhdan-Lusztig polynomials of matroids, extending previous results and applying valuation techniques to compute these polynomials for specific classes of matroids.

Contribution

It introduces an equivariant deletion formula for KL and Z-polynomials, and applies it to compute equivariant KL polynomials for certain matroids, including those of corank 2.

Findings

Derived an equivariant deletion formula relating KL and Z-polynomials.

Computed equivariant KL polynomials for matroids of corank 2.

Extended previous results to new classes of matroids.

Abstract

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.