Inverse Kazhdan-Lusztig polynomials of matroids under deletion

2025/10/01 by Braden, Tom, Ferroni, Luis, Matherne, Jacob P. +1 · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.01086

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.

Citations

• Log-concavity of inverse Kazhdan-Lusztig polynomials of paving matroids
• Chow functions for partially ordered sets
• Induced log-concavity of equivariant matroid invariants
• Valuations and the Hopf Monoid of Generalized Permutahedra
• Singular Hodge theory for combinatorial geometries
• Kazhdan-Lusztig polynomials of fan matroids, wheel matroids and whirl matroids
• The algebraic geometry of Kazhdan-Lusztig-Stanley polynomials
• Kazhdan-Lusztig polynomials of matroids: a survey of results and conjectures
• The Kazhdan-Lusztig polynomial of a matroid

Cited by

• Equivariant inverse Kazhdan--Lusztig polynomials of thagomizer matroids

Related

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• Kazhdan-Lusztig polynomials of matroids need not be unimodal
• On matroid modularity and the coefficients of the inverse Kazhdan-Lusztig polynomial of a matroid
• Equivariant inverse Kazhdan--Lusztig polynomials of thagomizer matroids