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 a family of elements called the Jucys—Murphy elements. A key ingredient in our analysis involves studying the character theory and center of the Hecke algebra, where many interesting questions and enigmas arise. One hope is that computational and machine learning tools can help elucidate these mysteries. Joint work with Jose Bastidas, Mathieu Guay-Paquet, Alejandro Morales, GaYee Park and Franco Saliola.