Factorizations in Hecke algebras I: Long‐cycle factorizations and Jucys–Murphy elements

Abstract

Given a permutation, there is a well‐developed literature studying the number of ways one can factor it into a product of other permutations subject to certain conditions. We initiate the analogous theory for the type‐ Iwahori–Hecke algebra by generalizing the notion of factorization in terms of the Jucys–Murphy elements. Some of the earliest and most foundational factorization results for the symmetric groups pertain to the long cycle. Our main results give ‐deformations of these long‐cycle factorizations and reveal ‐binomial, ‐Catalan, and ‐Narayana numbers along the way.