Quantum wreath products and Schur-Weyl duality I

Daniel K. Nakano, Chun-Ju Lai, Ziqing Xiang

Abstract: In this paper the authors introduce a new notion called the quantum wreath product, which produces an algebra

B w r_{Q} m a t h c a l H (d)

from a given algebra

B

, a positive integer

d

, and a choice

Q = (R, S, h o, s i g m a)

of parameters. Important examples 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

S i g m a_{m} w r S i g m a_{2}

between symmetric groups. In the first part of the paper, the authors develop a structure theory for the quantum wreath product algebras. 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.

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