# On fully commutative elements of type $\tilde B$ and $\tilde D$

## TL;DR

This paper classifies fully commutative elements in affine Coxeter groups of types B and D, constructs towers of these groups and their algebras, and proves faithfulness of the resulting algebraic structures.

## Contribution

It introduces a normal form for fully commutative elements in affine types B and D and establishes faithful towers of Coxeter groups and related algebras.

## Key findings

- Normal form classification for fully commutative elements
- Construction of faithful towers of Coxeter groups
- Faithfulness of affine Temperley-Lieb algebra towers

## Abstract

We define a tower of injections of $\tilde{B}$-type (resp. $\tilde{D}$-type) Coxeter groups $W(\tilde B_{n})$ (resp. $W(\tilde D_{n})$) for $n\geq 3$. Let $W^c(\tilde B_{n})$ (resp. $W^c(\tilde D_{n})$) be the set of fully commutative elements in $W(\tilde B_{n})$ (resp. $W(\tilde D_{n})$), we classify the elements of this set by giving a normal form for them. We define a $\tilde{B}$-type tower of Hecke algebras and we use the faithfulness at the Coxeter level to show that this last tower is a tower of injections. We use this normal form to define two injections from $W^c(\tilde B_{n-1})$ into $W^c(\tilde B_{n})$. We then define the tower of affine Temperley-Lieb algebras of type $\tilde{B }$ and use the injections above to prove the faithfulness of this tower. We follow the same track for $\tilde{D}$-type objects