**Speaker:** Sadek Alharbat (Paris)

**Title:** *Full Commutativity and Injectivity of Towers
of \(\tilde C\)-Type Structures*

**Abstract:**

We define a tower of injections of \(\tilde C\)-type Coxeter groups \(W(\tilde C_n)\) for \(n \geq 1\). We define a tower of Hecke algebras and show that it 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 faithfullness of this tower.