Factorizations in Hecke algebras I: long cycle factorizations and Jucys-Murphy elements

TL;DR

This paper extends the classical permutation factorization theory to type A Iwahori-Hecke algebras using Jucys-Murphy elements, introducing q-deformations of long cycle factorizations and uncovering related q-analog numbers.

Contribution

It introduces a novel framework for factorization in Hecke algebras, generalizing classical results and connecting them with q-analog combinatorial numbers.

Findings

q-deformations of long cycle factorizations derived

Identification of q-binomial, q-Catalan, and q-Narayana numbers

Extension of permutation factorization theory to Hecke algebras

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 A Iwahori-Hecke algebra by generalizing the notion of factorization in terms of the Jucys-Murphy elements. Some of the oldest and most foundational factorization results for the symmetric groups pertain to the long cycle. Our main results give q-deformations of these long cycle factorizations and reveal q-binomial, q-Catalan, and q-Narayana numbers along the way.