Tower of fully commutative elements of type A˜ and applications

Let Wc(A˜n) be the set of fully commutative elements in the affine Coxeter group W(A˜n) of type A˜. We classify the elements of Wc(A˜n) and give a normal form for them. 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 Wc(A˜n−1) into Wc(A˜n) and examine their properties. We finally consider the tower of affine Temperley–Lieb algebras of type A˜ and use the injections above to prove the injectivity of this tower.