2023/04/27 by Chun‐Ju Lai, Lai, Chun-Ju, Daniel K. Nakano +3 · 3 citations
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #FOS: Mathematics #Quantum Algebra (math.QA) #Quantum chaos and dynamical systems #Random Matrices and Applications #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2304.14181
openalex publication_date 2023/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper the authors introduce a new notion called the quantum wreath product, which is the algebra B \wrQ H(d) produced from a given algebra B, a positive integer d, and a choice Q=(R,S,ρ,σ) of parameters. Important examples that arise from our construction include many variants of the Hecke algebras, such as the Ariki-Koike algebras, the affine Hecke algebras and their degenerate version, Wan-Wang's wreath Hecke algebras, Rosso-Savage's (affine) Frobenius Hecke algebras, Kleshchev-Muth's affine zigzag algebras, and the Hu algebra that quantizes the wreath product Σm \wr Σ2 between symmetric groups. In the first part of the paper, the authors develop a structure theory for the quantum wreath products. Necessary and sufficient conditions for these algebras to afford a basis of suitable size are obtained. Furthermore, a Schur-Weyl duality is established via a splitting lemma and mild assumptions on the base algebra B. Our uniform approach encompasses many known results which were proved in a case by case manner. The second part of the paper involves the problem of constructing natural subalgebras of Hecke algebras that arise from wreath products. Moreover, a bar-invariant basis of the Hu algebra via an explicit formula for its extra generator is also described.