Quantum wreath products and Schur-Weyl duality I (Q6646835)

Let \(W\) be a Coxeter group and \(K\) a commutative ring. The Iwahori-Hecke algebra \(\mathcal{H}_{q}(W)\) is a \(q\)-deformation of the group algebra \(KW\) which was later extended to a deformation of a complex reflection group \(W\) (see [\textit{M. Broué} et al., J. Reine Angew. Math. 500, 127--190 (1998; Zbl 0921.20046)]).\N\NFor finite groups, there is a well-established theory involving subgroups and their relationship to the ambient group. However, for Hecke algebras, there are very few constructions of natural subalgebras that arise within these algebras. For example, an open problem is to develop a Sylow theory for Hecke algebra when \(q\) is specialized to a root of unity. A fundamental construction of Sylow subgroups for the symmetric group involves wreath products so a natural question to ask is whether one can make constructions of wreath products using Hecke algebras. The following problems will constitute the focus of this paper under review: (1) developing a theory of wreath products between (quantum) algebras that provides a uniform treatment to all Hecke-like algebras; (2) constructing ``the Hecke algebra'' \(\mathcal{H}_{m \wr d}\) that arises from the wreath product \(\Sigma_{m}\wr \Sigma_{d}\) between symmetric groups.\N\NIn 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. 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.