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Inverse Kazhdan-Lusztig polynomials of matroids under deletion

Tom Braden, Luis Ferroni, Jacob P. Matherne, Nutan Nepal

Abstract

We provide a deletion formula for the inverse Kazhdan--Lusztig polynomial and the inverse $Z$-polynomial of a matroid. Our formulas provide analogues to the deletion formulas of Braden--Vysogorets for Kazhdan--Lusztig and $Z$-polynomials. We discuss several consequences, which include closed formulas and recursions for these invariants on uniform matroids, projective geometries, glued cycles, and arbitrary matroids of corank $2$. As a relevant application of our deletion formula, we show the existence of a matroid of rank $19$ which disproves a conjecture of Xie and Zhang concerning a real-rootedness property of inverse Kazhdan--Lusztig polynomials.

Inverse Kazhdan-Lusztig polynomials of matroids under deletion

Abstract

We provide a deletion formula for the inverse Kazhdan--Lusztig polynomial and the inverse -polynomial of a matroid. Our formulas provide analogues to the deletion formulas of Braden--Vysogorets for Kazhdan--Lusztig and -polynomials. We discuss several consequences, which include closed formulas and recursions for these invariants on uniform matroids, projective geometries, glued cycles, and arbitrary matroids of corank . As a relevant application of our deletion formula, we show the existence of a matroid of rank which disproves a conjecture of Xie and Zhang concerning a real-rootedness property of inverse Kazhdan--Lusztig polynomials.

Paper Structure

This paper contains 15 sections, 14 theorems, 62 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Incidence algebras and kernels
  5. A recapitulation on the Kazhdan--Lusztig invariants
  6. Proof of the deletion formulas
  7. The Kazhdan--Lusztig basis
  8. A homomorphism for single element deletions
  9. The deletion formula for inverse KL polynomials
  10. Deletion formula for the inverse Z-polynomial
  11. Some special cases and examples
  12. Uniform matroids
  13. Glued cycles
  14. Single element deletions of projective geometries
  15. A counterexample to Conjecture \ref{['conj:xie-zhang']}

Key Result

Theorem 1.1

Let $\mathsf{M}$ be a loopless matroid of rank $k$, and let $i\in E$ be an element of the ground set that is not a coloop. Then,

Figures (2)

  • Figure 1: The graphic matroid $\mathsf{C}_{5,6}$.
  • Figure 2: The edges on the left are labelled by $\{1,5,6,8\}$, the edges on the right by $\{3,4,7\}$ and the bottom one by $\{2\}$.

Theorems & Definitions (29)

  • Theorem 1.1: braden-vysogorets
  • Conjecture 1.2
  • Conjecture 1.3: xie-zhang-conjecture
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1: proudfoot-xu-youngbraden-vysogorets
  • Theorem 2.2: braden-huh-matherne-proudfoot-wangferroni-matherne-stevens-vecchigao-ruan-xie
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • ...and 19 more