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

2025/06/10 by Jose Bastidas, Bastidas, Jose, Sarah Brauner +9
Mathematics · #05A05 #05A15 #05A30 #05E05 #20C08 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2506.08883

openalex publication_date 2025/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

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Related

• Long Cycle Factorizations : Bijective Computation in the General Case
• Bijections and symmetries for the factorizations of the long cycle
• James' Conjecture for Hecke algebras of exceptional type, I
• Products of Geck-Rouquier conjugacy classes and the Hecke algebra of composed permutations
• A generalization of Deodhar's defect statistic for Iwahori--Hecke algebras of type BC