Star and weak star irreducible fully commutative elements in Coxeter groups of affine types $\widetilde{B}$ and $\widetilde{D}$

TL;DR

This paper classifies star and weak star irreducible fully commutative elements in affine Coxeter groups of types B and D, and applies this to prove faithfulness of a diagrammatic algebra representation and describe Lusztig's -function.

Contribution

It provides a complete classification of certain irreducible elements in affine Coxeter groups and applies this to representation theory and Lusztig's -function.

Findings

Classification of star and weak star irreducible elements in B and D types

New proof of faithfulness of diagrammatic Temperley-Lieb algebra representation

Explicit description of Lusztig's -function

Abstract

The star operation, originally introduced by Kazhdan and Lusztig, was later specialized by Ernst to the so-called weak star reduction on the set of fully commutative elements of a Coxeter group. In this paper, we classify the star and weak star irreducible fully commutative elements in Coxeter groups of affine types $\widetilde{B}_{n+1}$ and $\widetilde{D}_{n+2}$. Focusing then on the case of type $\widetilde{D}_{n+2}$, we use the classification of star irreducible elements to provide a new proof of the faithfulness of a diagrammatic representation of the corresponding generalized Temperley-Lieb algebra, along with an explicit description of Lusztig's $\mathbf{a}$-function.