Non-cancellable elements in type affine $C$ Coxeter groups

Abstract

Let $(W,S)$ be a Coxeter system and suppose that $w\in W$ is fully commutative (in the sense of Stembridge) and has a reduced expression beginning (respectively, ending) with $s\in S$. If there exists $t\in S$ such that $s$ and $t$ do not commute and $tw$ (respectively, $wt$) is no longer fully commutative, we say that $w$ is left (respectively, right) weak star reducible by $s$ with respect to $t$. In this paper, we classify the fully commutative elements in Coxeter groups of types $B$ and affine $C$ that are irreducible under weak star reductions. In a sequel to this paper, the classification of the weak star irreducible elements in a Coxeter system of type affine $C$ will provide the groundwork for inductive arguments used to prove the faithfulness of a generalized Temperley--Lieb algebra of type affine $C$ by a particular diagram algebra.

The statement of Proposition 3.2.3 contained a small error in the previous version. This has been addressed in the new version. Thankfully, this error had no impact on the remainder of the paper. The published version in Int. Electron. J. Algebra, 8:191-218, 2010 still contains the error. 21 pages, 22 figures

References in corpus (1)

• A diagrammatic representation of an affine $C$ Temperley--Lieb algebra

Cited by in corpus (3)

• Diagram calculus for a type affine $C$ Temperley--Lieb algebra, I
• Conjugacy classes of cyclically fully commutative elements in Coxeter groups of type A
• A cellular quotient of the Temperley--Lieb algebra of type $D$