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

## TL;DR

This paper classifies certain fully commutative elements in affine type C Coxeter groups that cannot be simplified via weak star reductions, aiding the understanding of related algebraic structures.

## Contribution

It provides a classification of weak star irreducible fully commutative elements in types B and affine C Coxeter groups, advancing the structural understanding of these groups.

## Key findings

- Identified all weak star irreducible elements in types B and affine C.
- Established foundational results for inductive proofs in algebraic representations.
- Facilitated future work on faithfulness of generalized Temperley--Lieb algebras.

## 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.