Atomic decomposition for an affine Weyl group of type $G_2$

TL;DR

This paper demonstrates that elements of the Kazhdan--Lusztig basis for the spherical Hecke algebra of type G2 can be decomposed atomically, leading to a new algorithm for computing generalized Kostka--Foulkes polynomials in this type.

Contribution

It introduces an atomic decomposition for the Kazhdan--Lusztig basis in type G2 and provides a novel algorithm for calculating related polynomials.

Findings

Atomic decomposition of basis elements established

New algorithm for Kostka--Foulkes polynomial computation

Enhanced understanding of algebraic structures in type G2

Abstract

We show that the elements of the Kazhdan--Lusztig basis of the spherical Hecke algebra of type $G_2$ have an atomic decomposition. As a by-product, we obtain a new algorithm to compute generalized Kostka--Foulkes polynomials in type $G_2$.