The Hecke algebra is in general not quasi-hereditary, meaning that its module category is not a highest weight category; while it admits a quasi-hereditary cover by the category 𝒪 of a certain rational Cherednik algebra due to Ginzburg-Guay-Opdam-Rouquier. It was later shown in type A that this category 𝒪 can be realized concretely as the module category of Dipper-James’s q-Schur algebra, but this realization problem remains open beyond types A and B. An essential step for type D, i.e., the complex reflection group G(2,2,n), is to study Hu’s Hecke subalgebra, which deforms a wreath product that is not a Coxeter group. In this talk, I’ll introduce a new theory allowing us to take the wreath product of an algebra by a Hecke algebra. Before our work, wreath products related to Hecke algebras were worked out at the degenerate level by Wan-Wang. Our wreath product produces the Ariki-Koike algebras as special cases as well as new ‘Hecke algebras’ of wreath products between symmetric groups. These are the first steps towards answering the realization problem for complex reflection groups.
This is joint work with D. Nakano and Z. Xiang.
This video is part of the University of Georgia‘s Algebra seminar.
