On Cellularity of Hecke Algebras for Wreath Products

Abstract

The (generalized) Hu algebra is a nontrivial quantization of the wreath product $Σ_m \wr Σ_d$ between symmetric groups, whose representation theory controls the Hecke algebra of the complex reflection group $G(d,d,md)$. In this paper, we construct a unified basis for this algebra and establish its cellular algebra structure in the case $d = 2$. As an application, our construction provides an elementary realization of the simple modules for the Hecke algebra of type $D_{2m}$ that are parameterized by bipartitions of size $(m,m)$.