The algebraic geometry of Kazhdan-Lusztig-Stanley polynomials

TL;DR

This paper explores the algebraic and geometric structures underlying Kazhdan-Lusztig-Stanley polynomials, connecting combinatorial, algebraic, and geometric perspectives across various mathematical objects.

Contribution

It unifies cohomological interpretations of Kazhdan-Lusztig-Stanley polynomials within a single geometric framework, extending known results to broader contexts.

Findings

Cohomological interpretations for Weyl groups, polytopes, and matroids.

Unified geometric framework for these polynomials.

Connections between combinatorics, algebra, and geometry.

Abstract

Kazhdan-Lusztig-Stanley polynomials are a combinatorial generalization of Kazhdan-Lusztig polynomials of for Coxeter groups that include g-polynomials of polytopes and Kazhdan-Lusztig polynomials of matroids. In the cases of Weyl groups, rational polytopes, and realizable matroids, one can count points over finite fields on flag varieties, toric varieties, or reciprocal planes to obtain cohomological interpretations of these polynomials. We survey these results and unite them under a single geometric framework.