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

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.