Affine Wreath Product Algebras

Abstract

We study the structure and representation theory of affine wreath product algebras and their cyclotomic quotients. These algebras, which appear naturally in Heisenberg categorification, simultaneously unify and generalize many important algebras appearing in the literature. In particular, special cases include degenerate affine Hecke algebras, affine Sergeev algebras (degenerate affine Hecke-Clifford algebras), and wreath Hecke algebras. In some cases, specializing the results of the current paper recovers known results, but with unified and simplified proofs. In other cases, we obtain new results, including proofs of two open conjectures of Kleshchev and Muth.

Degenerate cyclotomic Hecke algebras and higher level Heisenberg categorification

“…These algebras have been studied in [Sava], where they are called affine wreath product algebras, and denoted A n (B). In particular, cyclotomic quotients A C n (B) of these algebras are defined [Sava,§6], and it is shown that level one cyclotomic quotients are isomorphic to B ⊗n ⋊ S n (see [Sava,Cor. 6.13]).…”

Frobenius Heisenberg categorification

“…which is the identity by (18). Composing in the other order, we obtain a (1+k dim F )× (1 + k dim F ) matrix.…”

“…Moreover, Heis F,k can be presented equivalently as the strict k-linear monoidal supercategory generated by the objects Q + , Q − , and morphisms s, x, c, d, c , d , and β f , f ∈ F , subject only to the relations (4), (6) to (10), (13), (14), and (18) to (23). In the above relations, in addition to the rightward crossing t defined by (3), we have used the left crossing t : 5 (2019) and the negatively dotted bubbles defined, for f ∈ F , by…”

“…The definition of the supercategory Heis F,k is inspired by the affine wreath product algebras studied in [18]. One benefit of the inversion relation presentation of Definition 1.1 is that it makes it relatively straightforward to verify that Heis F,k acts naturally on suitable categories.…”

“…In this case, the morphism x (corresponding diagrammatically to the dot) would be odd and one of the left adjunction relations (30) and (31) would involve a sign. In a different direction, one can define natural q-deformations of the affine wreath product algebras of [18]; this is done in [17]. This leads to a q-deformation of the supercategory Heis F,k [6].…”