Abstract
We define a tower of injections of B˜-type (resp. D˜-type) Coxeter groups W(B˜n) (resp. W(D˜n)). Let Wc(B˜n) (resp. Wc(D˜n)) be the set of fully commutative elements in W(B˜n) (resp. W(D˜n)), we classify the elements of this set by giving a normal form for them. We define a B˜-type tower of Hecke algebras and we use the faithfulness at the Coxeter level to show that this last tower is a tower of injections. We use this normal form to define two injections from Wc(B˜n−1) into Wc(B˜n). We then define the tower of affine Temperley-Lieb algebras of type B˜ and use the injections above to prove the faithfulness of this tower. We follow the same track for D˜-type objects.
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