Mask formulas for cograssmannian Kazhdan-Lusztig polynomials

TL;DR

This paper introduces two novel methods for constructing mask sets on cograssmannian permutations, enhancing the understanding and computation of Kazhdan-Lusztig polynomials via Deodhar's formula.

Contribution

It provides two new constructions based on Lascoux-Schutzenberger's formula and Zelevinsky's geometric interpretation, addressing a question posed by Deodhar.

Findings

Two distinct mask constructions for cograssmannian permutations

Connections to Bruhat order and Bott-Samelson resolution

Improved understanding of Kazhdan-Lusztig polynomials

Abstract

We give two contructions of sets of masks on cograssmannian permutations that can be used in Deodhar's formula for Kazhdan-Lusztig basis elements of the Iwahori-Hecke algebra. The constructions are respectively based on a formula of Lascoux-Schutzenberger and its geometric interpretation by Zelevinsky. The first construction relies on a basis of the Hecke algebra constructed from principal lower order ideals in Bruhat order and a translation of this basis into sets of masks. The second construction relies on an interpretation of masks as cells of the Bott-Samelson resolution. These constructions give distinct answers to a question of Deodhar.