A fully commutative element of a Coxeter group is one for which any reduced word can be obtained from another by applying only commutation relations. For example, in a type A Coxeter group, these are exactly the 321-avoiding permutations, known to be counted by the Catalan numbers. For other types, these fully commutative elements have been studied and enumerated by John Stembridge, and have been found to be related to Temperley-Lieb algebras and Khovanov-Lauda-Rouquier algebras.
In this talk, we’ll extend the theory to more general complex reflection groups. We’ll deal with some difficulties in generalizing the notion of full commutativity and see some interesting combinatorial structures, including the related Catalan triangle.