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.