Séminaire Lotharingien de Combinatoire, 95B.22 (2026), 12 pp.
Jose Bastidas, Sarah Brauner, Mathieu Guay-Paquet, Alejandro H. Morales, GaYee Park and Franco Saliola
Factorizations for the Iwahori-Hecke Algebra
Abstract.
We initiate the theory of factorizations in the Type A Iwahori-Hecke algebra, and prove q-analogues of the famous Harer-Zagier formula, as well as generalizations of classical factorization problems for the long-cycle due to Bédard, Goulden, Goupil, Hurwitz, Jackson, Matsumoto, Novak, and others. Our work reframes the notion of factorization in terms of certain structure constants in the Hecke algebra center, and provides a uniform method of approaching such problems. Our formulas reveal significant combinatorial structure in the Hecke algebra center, including the appearance of q-binomial, q-Catalan, and q-Narayana numbers.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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