Star and weak star irreducible fully commutative elements in Coxeter groups of affine types B and D

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 B~n+1 and D~n+2. Focusing then on the case of type 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 a-function.