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Kazhdan-Lusztig polynomials of braid matroids

Luis Ferroni, Matt Larson

Abstract

We provide a combinatorial interpretation of the Kazhdan--Lusztig polynomial of the matroid arising from the braid arrangement of type $\mathrm{A}_{n-1}$, which gives an interpretation of the intersection cohomology Betti numbers of the reciprocal plane of the braid arrangement. Moreover, we prove an equivariant version of this result. The key combinatorial object is a class of matroids arising from series-parallel networks. As a consequence, we prove a conjecture of Elias, Proudfoot, and Wakefield on the top coefficient of Kazhdan--Lusztig polynomials of braid matroids, and we provide explicit generating functions for their Kazhdan--Lusztig and $Z$-polynomials.

Kazhdan-Lusztig polynomials of braid matroids

Abstract

We provide a combinatorial interpretation of the Kazhdan--Lusztig polynomial of the matroid arising from the braid arrangement of type , which gives an interpretation of the intersection cohomology Betti numbers of the reciprocal plane of the braid arrangement. Moreover, we prove an equivariant version of this result. The key combinatorial object is a class of matroids arising from series-parallel networks. As a consequence, we prove a conjecture of Elias, Proudfoot, and Wakefield on the top coefficient of Kazhdan--Lusztig polynomials of braid matroids, and we provide explicit generating functions for their Kazhdan--Lusztig and -polynomials.
Paper Structure (10 sections, 13 theorems, 18 equations, 4 figures, 2 tables)

This paper contains 10 sections, 13 theorems, 18 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Overview
  3. Kazhdan--Lusztig polynomials of matroids
  4. The Kazhdan--Lusztig theory of braid matroids
  5. Main results
  6. Background
  7. Series-parallel matroids
  8. Quasi series-parallel matroids
  9. Simple quasi series-parallel matroids
  10. Proof of the main results

Key Result

Theorem 1.1

Let $\mathcal{A}(n,r)$ denote the set of all quasi series-parallel matroids on $[n]$ of rank $r$ and, let $\mathcal{S}(n,r)$ denote the set of simple quasi series-parallel matroids. Then

Figures (4)

  • Figure 1: Isomorphism classes of series-parallel matroids on $[6]$ of rank $3$.
  • Figure 2: Isomorphism classes of quasi series-parallel matroids on $[4]$ of rank $2$.
  • Figure 3: Isomorphism classes of simple quasi series-parallel matroids on $[7]$ of rank $4$.
  • Figure 4: Two simple series-parallel matroids of rank $4$ on $[7]$ and their corresponding triangular cacti.

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1: brylawski
  • Example 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Definition 2.5
  • Proposition 2.6
  • Example 2.7
  • ...and 16 more