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

We present a combinatorial formula using skew Young tableaux for the coefficients of Kazhdan-Lusztig polynomials for sparse paving matroids. These matroids are known to be logarithmically almost all matroids, but are conjectured to be almost all matroids. We also show the positivity of these coefficients using our formula. In special cases, such as uniform matroids, our formula has a nice combinatorial interpretation.

The Leading Coefficients for an Affine Weyl Group of Type G 2 ˜ : The Lowest Two-Sided Cell Case

This study focusses on the leading coefficients μ u , w of the Kazhdan–Lusztig polynomials P u , w for the lowest cell c 0 of an affine Weyl group of type G 2 ˜ and gives an estimation μ u , w ≤ 3 for u , w ∈ c 0 .

A Combinatorial Formula for Kazhdan-Lusztig Polynomials of $\rho$-Removed Uniform Matroids

Let $\rho$ be a non-negative integer. A $\rho$-removed uniform matroid is a matroid obtained from a uniform matroid by removing a collection of $\rho$ disjoint bases. We present a combinatorial formula for Kazhdan–Lusztig polynomials of $\rho$-removed uniform matroids, using skew Young Tableaux. Even for uniform matroids, our formula is new, gives manifestly positive integer coefficients, and is more manageable than known formulas.

Kazhdan-Lusztig Polynomials of Matroids Under Deletion

We present a formula which relates the Kazhdan–Lusztig polynomial of a matroid $M$, as defined by Elias, Proudfoot and Wakefield, to the Kazhdan–Lusztig polynomials of the matroid obtained by deleting an element, and various contractions and localizations of $M$. We give a number of applications of our formula to Kazhdan–Lusztig polynomials of graphic matroids, including a simple formula for the Kazhdan–Lusztig polynomial of a parallel connection graph.

A Combinatorial Formula for the Coefficient of q in Kazhdan–Lusztig Polynomials

We propose a combinatorial interpretation of the coefficient of $q$ in Kazhdan–Lusztig polynomials and we prove it for finite simply-laced Weyl groups.