The algebraic geometry of Kazhdan–Lusztig–Stanley polynomials

Abstract

Kazhdan–Lusztig–Stanley polynomials are combinatorial generalizations of Kazhdan–Lusztig polynomials of 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.