A Combinatorial Formula for Kazhdan-Lusztig Polynomials of Sparse Paving Matroids

TL;DR

This paper proves the positivity of Kazhdan-Lusztig polynomials for sparse paving matroids using a new combinatorial formula involving skew Young tableaux, supporting the broader conjecture for all matroids.

Contribution

It introduces a simple combinatorial formula for Kazhdan-Lusztig polynomials of sparse paving matroids, advancing understanding of their positivity.

Findings

Positivity of Kazhdan-Lusztig polynomials for sparse paving matroids established.

New combinatorial formula involving skew Young tableaux derived.

Supports the conjecture that all matroids have non-negative Kazhdan-Lusztig polynomial coefficients.

Abstract

We prove the positivity of Kazhdan-Lusztig polynomials for sparse paving matroids, which are known to be logarithmically almost all matroids, but are conjectured to be almost all matroids. The positivity follows from a remarkably simple combinatorial formula we discovered for these polynomials using skew young tableaux. This supports the conjecture that Kazhdan-Lusztig polynomials for all matroids have non-negative coeffiecients. In special cases, such as uniform matroids, our formula has a nice combinatorial interpretation.