On the fully commutative elements of type $\tilde{C}$ and faithfulness of related towers

Abstract

We define a tower of injections of $\tilde{C}$-type Coxeter groups $W({\tilde{C}}_n)$ for $n\ge 1$. We define a tower of Hecke algebras and we use the faithfulness at the Coxeter level to show that this last tower is a tower of injections. Let $W^c({\tilde{C}}_n)$ be the set of fully commutative elements in $W({\tilde{C}}_n)$, we classify the elements of $W^c({\tilde{C}}_n)$ and give a normal form for them. We use this normal form to define two injections from $W^c({\tilde{C}}_{n-1})$ into $W^c({\tilde{C}}_n)$. We then define the tower of affine Temperley-Lieb algebras of type $\tilde{C}$ and use the injections above to prove the faithfulness of this tower.

arXiv comment24 pages. Version 2 includes some corrections and improved notation