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Deletion formulas for equivariant Kazhdan-Lusztig polynomials of matroids

Luis Ferroni, Jacob P. Matherne, Lorenzo Vecchi

TL;DR

The paper develops an equivariant deletion framework for Kazhdan–Lusztig and Z-polynomials of matroids, yielding a precise recursion that relates $P_M^W(x)$ and $Z_M^W(x)$ to deletions and contractions along with orbit-summed correction terms. It demonstrates that equivariant $Z$-positivity need not hold in general, and uses the deletion formula to compute the equivariant KL polynomial for graphic matroids formed by gluing two cycles, employing Littlewood–Richardson coefficients for explicit representation-theoretic decompositions. The authors further leverage matroid valuations in the corank-2 setting to provide a structured formula expressing $P_M^W(x)$ in terms of uniform corank-1 and corank-2 components, enabling concrete computations in this class. Overall, the work extends non-equivariant deletion recurrences to the equivariant realm, enabling both theoretical understanding and practical calculations of equivariant KL and Z-polynomials via deletions, gluings, and valuations.

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 $γ$-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.

Deletion formulas for equivariant Kazhdan-Lusztig polynomials of matroids

TL;DR

The paper develops an equivariant deletion framework for Kazhdan–Lusztig and Z-polynomials of matroids, yielding a precise recursion that relates and to deletions and contractions along with orbit-summed correction terms. It demonstrates that equivariant -positivity need not hold in general, and uses the deletion formula to compute the equivariant KL polynomial for graphic matroids formed by gluing two cycles, employing Littlewood–Richardson coefficients for explicit representation-theoretic decompositions. The authors further leverage matroid valuations in the corank-2 setting to provide a structured formula expressing in terms of uniform corank-1 and corank-2 components, enabling concrete computations in this class. Overall, the work extends non-equivariant deletion recurrences to the equivariant realm, enabling both theoretical understanding and practical calculations of equivariant KL and Z-polynomials via deletions, gluings, and valuations.

Abstract

We study equivariant Kazhdan--Lusztig (KL) and -polynomials of matroids. We formulate an equivariant generalization of a result by Braden and Vysogorets that relates the equivariant KL and -polynomials of a matroid with those of a single-element deletion. We also discuss the failure of equivariant -positivity for the -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~ via valuations. This provides an application of the machinery of Elias, Miyata, Proudfoot, and Vecchi to corank matroids, and it extends results of Ferroni and Schröter.
Paper Structure (13 sections, 9 theorems, 40 equations, 2 figures)

This paper contains 13 sections, 9 theorems, 40 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Preliminaries
  4. Equivariant Kazhdan--Lusztig and Z-polynomials
  5. Equivariant gamma-positivity
  6. Equivariant deletion formula
  7. Applications: gluing of cycles and corank 2 matroids
  8. Equivariant KL polynomial of two glued cycles
  9. Representation theory generalities
  10. Application to gluing cycles
  11. Corank 2 matroids.
  12. Equivariant valuativity for Kazhdan--Lusztig polynomials.
  13. Formula for corank 2 matroids.

Key Result

Theorem 1.1

Let $\mathsf{M}$ be a loopless matroid of rank $k$ on $E$, and let $i\in E$ be an element that is not a coloop. Then, where $W_F$ is the stabilizer of the flat $F$, and $\mathscr{S}_i/W_i$ is the quotient of $\mathscr{S}_i$ by the group action $W_i$.

Figures (2)

  • Figure 1: $\uplambda = [6,2^7]$ and $\mu=[3,2^2,1^3]$.
  • Figure 2: The unique filling in each of the four cases

Theorems & Definitions (23)

  • Theorem 1.1
  • Remark 2.2
  • Theorem 2.3: gedeon-proudfoot-young-equivariant
  • Definition 2.4
  • Example 2.5
  • Lemma 2.6
  • proof
  • Proposition 2.7
  • proof
  • proof : Proof of Theorem \ref{['thm:equivariant-deletion-formula']}
  • ...and 13 more