The explicit bijection between fully commutative elements and non-crossing diagrams

Abstract

Let Wc(An) be the set of fully commutative elements in the Coxeter group of type An. We determine explicitly the unique bijection between Wc(An) and the set of non-crossing diagrams of n + 1 strands that is compatible with the product of monomials in the Temperley-Lieb algebra of type An on one side, and the concatenation of diagrams on the other. We describe algorithms implementing this bijection and its inverse.