Kazhdan-Lusztig polynomials of thagomizer matroids

TL;DR

This paper introduces thagomizer matroids and calculates their Kazhdan-Lusztig polynomials, revealing a combinatorial interpretation involving Dyck paths and proposing a related conjecture for the equivariant case.

Contribution

It defines thagomizer matroids and provides an explicit formula for their Kazhdan-Lusztig polynomials, linking algebraic invariants to Dyck path combinatorics.

Findings

Coefficient of t^k equals number of Dyck paths with k long ascents

Explicit polynomial formulas for thagomizer matroids

Conjecture on S_n-equivariant Kazhdan-Lusztig polynomial

Abstract

We introduce thagomizer matroids and compute the Kazhdan-Lusztig polynomial of a rank $n+1$ thagomizer matroid by showing that the coefficient of $t^k$ is equal to the number of Dyck paths of semilength $n$ with $k$ long ascents. We also give a conjecture for the $S_n$-equivariant Kazhdan-Lusztig polynomial of a thagomizer matroid.