Quantum wreath products and Schur-Weyl duality I

In this paper the authors introduce a new notion called the quantum wreath product, which is the algebra \(B \wr_Q \mathcal{H}(d)\) produced from a given algebra \(B\), a positive integer \(d\), and a choice \(Q=(R,S,\rho,\sigma)\) 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 \(\Sigma_m \wr \Sigma_2\) 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.