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.

arXiv comment27p. Includes some corrections and improved notation. The normal form for affine fully commutative elements of type $\tilde{A}$ and for fully commutative affine braids was established in arXiv:1311.7089 "A classification of affine fully commutative elements". We added a new application to the injectivity of the tower of affine Temperley-Lieb algebras of type $tildeA$