Tower of fully commutative elements of type $\tilde A$ and applications

TL;DR

This paper classifies fully commutative elements in affine Coxeter groups of type A, provides a normal form, and applies it to affine braids and Temperley-Lieb algebras, establishing injectivity in the tower.

Contribution

It introduces a classification and normal form for fully commutative affine A elements, enabling new applications and proofs in algebraic structures.

Findings

Classification and normal form for A affine fully commutative elements

Construction of injections between A affine Coxeter groups

Proof of injectivity of the affine Temperley-Lieb algebra tower

Abstract

Let $W^c(\tilde A_{n})$ be the set of fully commutative elements in the affine Coxeter group $W(\tilde A_{n})$ of type $\tilde{A}$. We classify the elements of $W^c(\tilde A_{n})$ and give a normal form for its elements. We give a first application of this normal form to fully commutative affine braids. We then use this normal form to define two injections from $W^c(\tilde A_{n-1})$ into $W^c(\tilde A_{n})$ and examine their properties. We then consider the tower of affine Temperley-Lieb algebras of type $\tilde{A }$ and use the injections above to prove the injectivity of this tower.