Kazhdan-Lusztig polynomials for $\tilde{B}_2$

TL;DR

This paper explicitly computes Kazhdan-Lusztig polynomials for the affine Coxeter system of type B_2, addressing a longstanding challenge in algebraic combinatorics.

Contribution

It provides the first explicit calculations of Kazhdan-Lusztig polynomials for the B_2 affine Coxeter system, advancing understanding of their structure.

Findings

Explicit formulas for B_2 Kazhdan-Lusztig polynomials

New computational methods for affine Coxeter systems

Enhanced understanding of algebraic combinatorics in affine types

Abstract

Kazhdan and Lusztig define, for an arbitrary Coxeter system $(W,S)$, a family of polynomials indexed by pairs of elements of $W$. Despite their relevance and elementary definition, the explicit computation of these polynomials is still one of the hardest open problems in algebraic combinatorics. In this paper we explicitly compute Kazhdan-Lusztig polynomials for a Coxeter system of type $\tilde{B}_2$.