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

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 ${\tilde{B}}_{n+1}$ and ${\tilde{D}}_{n+2}$. Focusing then on the case of type ${\tilde{D}}_{n+2}$, 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.