A quantum wreath product is the algebra produced from a given (not necessarily commutative) algebra B, a positive integer d, and a choice of certain coefficients in B ⊗ B. Important examples include variants of the Hecke algebras, such as (1) affine Hecke algebras and their degenerate version, (2) Wan-Wang’s wreath Hecke algebras, (3) Kleshchev-Muth’s affinization algebras, (4) Rosso-Savage’s (affine) Frobenius Hecke algebras, (5) endomorphism algebras arising from Elias’s Hecke-type categories, (6) Mathas-Stroppel’s Rees affine Frobenius Hecke algebras, and (7) Hu algebra, which quantizes the wreath product Sm ≀ S2 between the symmetric groups. Our goal is to develop a uniform approach to the structure and representation theory in order to encompass known results which were proved in a case by case manner. In this talk, I’ll focus on the Schur-Weyl duality and the Clifford theory. Our theory is motivated by (and has application to) the Ginzburg-Guay-Opdam-Rouquier problem on quasi-hereditary covers of Hecke algebras for complex reflection groups.
This is a joint work with Dan Nakano and Ziqing Xiang.
This video is part of the conference Representation Theory and Geometry that took place at the University of Georgia.
