A generalization of Deodhar's defect statistic for Iwahori--Hecke algebras of type $BC$

TL;DR

This paper extends Deodhar's defect statistic to type BC Coxeter groups, providing a combinatorial formula for expanding products of Kazhdan--Lusztig basis elements indexed by smooth hyperoctahedral group elements.

Contribution

It introduces a type-BC extension of Deodhar's defect statistic, generalizing previous type-A results to hyperoctahedral groups and their Kazhdan--Lusztig basis expansions.

Findings

Provides a combinatorial description for type-BC Kazhdan--Lusztig basis products.

Extends Deodhar's defect statistic to hyperoctahedral groups.

Enables explicit calculations of basis expansions in type BC.

Abstract

Let $H$ be the Iwahori--Hecke algebra corresponding to any Coxeter group. Deodhar's defect statistic [Geom. Dedicata 36, (1990) pp.95--119] allows one to expand products of simple Kazhdan--Lusztig basis elements of $H$ in the natural basis of $H$. Clearwater and the third author gave a type-$A$ extension [Ann. Comb. 25, no. 3 (2021) pp.757--787] of this formula which combinatorially describes the natural expansion of products of Kazhdan--Lusztig basis elements indexed by smooth elements of the symmetric group. We similarly give a type-$BC$ extension of Deodhar's result which combinatorially describes the natural expansion of Kazhdan--Lusztig basis elements indexed by hyperoctahedral group elements which are simultaneously smooth in types $B$ and $C$.