University of Sydney Algebra Seminar

Chun-Ju Lai

Friday 4 September, 12-1pm, in Carslaw 275

Foundations and Applications of Quantum Wreath Products

Roughly speaking, a quantum wreath product is a deformation algebra of the classical wreath product of a monoid (or group) by the symmetric group, where the relation coefficients are in a tensor product algebra rather than in the ground field. From this perspective, algebras admitting a Bernstein-Lusztig presentation are quantum wreath products. This framework encompasses (1) Vigneras' pro-p Hecke algebras and their metaplectic analogs, (2) Kleshchev and Muth's affine zigzag algebras, and (3) the ``Hecke algebra'' for the wreath products between symmetric groups arising from the work of Jun Hu. I'll speak about recent progress in seemingly unrelated directions using general theory of quantum wreath products. They are (A) realization problem of imaginary strata of affine Khovanov-Lauda-Rouquier algebras in positive characteristic, (B) failure of Local Shimura correspondence at the pro-p level, (C) the Ginzburg-Guay-Opdam-Rouquier conjecture on constructing highest weight covers for non-semisimple Hecke algebras of complex reflection groups, and (D) categorification perspective of towers of quantum wreath products. The talk is based on joint work with Buciumas (Pohang), Cailan Li (Academia Sinica), Minets (Bonn), Nakano (Georgia), and Xiang (Shenzhen).