The inverse Kazhdan-Lusztig polynomial of a matroid

Abstract

In analogy with the classical Kazhdan-Lusztig polynomials for Coxeter groups, Elias, Proudfoot and Wakefield introduced the concept of Kazhdan-Lusztig polynomials for matroids. It is known that both the classical Kazhdan-Lusztig polynomials and the matroid Kazhdan-Lusztig polynomials can be considered as special cases of the Kazhdan-Lusztig-Stanley polynomials for locally finite posets. In the framework of Kazhdan-Lusztig-Stanley polynomials, we study the inverse of Kazhdan-Lusztig-Stanley functions and define the inverse Kazhdan-Lusztig polynomials for matroids. We also compute these polynomials for boolean matroids and uniform matroids. As an unexpected application of the inverse Kazhdan-Lusztig polynomials, we obtain a new formula to compute the Kazhdan-Lusztig polynomials for uniform matroids. Similar to the Kazhdan-Lusztig polynomial of a matroid, we conjecture that the coefficients of its inverse Kazhdan-Lusztig polynomial are nonnegative and log-concave.

17 pages

References in corpus (2)

• Kazhdan-Lusztig polynomials of matroids: a survey of results and conjectures
• A semi-small decomposition of the Chow ring of a matroid

Cited by in corpus (6)

• Hilbert-Poincaré series of matroid Chow rings and intersection cohomology
• Stressed hyperplanes and Kazhdan-Lusztig gamma-positivity for matroids
• Matroid relaxations and Kazhdan-Lusztig non-degeneracy
• Kazhdan-Lusztig polynomials of braid matroids
• Deletion formulas for equivariant Kazhdan-Lusztig polynomials of matroids
• On matroid modularity and the coefficients of the inverse Kazhdan-Lusztig polynomial of a matroid