Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Bárbara Muniz, David Plaza, Claudia Rojas-andías

TL;DR

This work establishes that, for the affine Weyl group of type $G_2$, the spherical Kazhdan–Lusztig basis elements admit atomic decompositions with nonnegative coefficients in the atomic basis. It delivers two proofs: (i) a detailed inverse step-by-step decomposition using auxiliary $M^A_ u$ elements, and (ii) a modified, positivity-preserving construction of adjusted pre-canonical bases that directly yield positive atomic expansions. As a by-product, the authors provide a concrete algorithm to compute generalized Kostka–Foulkes polynomials in type $G_2$ and supply a SageMath implementation. The results advance understanding of atomicity beyond type $A$, with implications for crystallographic realizations and combinatorial descriptions of $q$-weight multiplicities in non-simply-laced types.

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 $G_2$.

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

TL;DR

This work establishes that, for the affine Weyl group of type , the spherical Kazhdan–Lusztig basis elements admit atomic decompositions with nonnegative coefficients in the atomic basis. It delivers two proofs: (i) a detailed inverse step-by-step decomposition using auxiliary elements, and (ii) a modified, positivity-preserving construction of adjusted pre-canonical bases that directly yield positive atomic expansions. As a by-product, the authors provide a concrete algorithm to compute generalized Kostka–Foulkes polynomials in type and supply a SageMath implementation. The results advance understanding of atomicity beyond type , with implications for crystallographic realizations and combinatorial descriptions of -weight multiplicities in non-simply-laced types.

Abstract

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

Paper Structure

This paper contains 10 sections, 9 theorems, 58 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The affine Weyl group of type
  4. Pre-canonical bases
  5. First approach
  6. Inverse step-by-step decompositions
  7. Step-by-step decompositions
  8. Atomic Decomposition
  9. Second approach
  10. Computing Kostka--Foulkes polynomials

Key Result

Theorem A

For every dominant weight $\lambda$ we have an expansion where $\leq$ denotes the dominance order on weights and the polynomials $a_{\lambda,\mu}(q)$ have non-negative coefficients. In other words, in type $G_2$, all the Kazhdan--Lusztig basis elements associated to spherical elements admits an atomic decomposition.

Figures (2)

  • Figure 1: A root system of type $G_2$
  • Figure 2: Type $G_2$ alcoves. The dots correspond to $X$. The dominant chamber is highlighted. The fundamental alcove is colored in light blue. The green triangle correspond to $\theta$-elements.

Theorems & Definitions (33)

  • Theorem A
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 3.1
  • Lemma 3.2
  • proof
  • ...and 23 more