Affine wreath product algebras with trace maps of generic parity

Abstract

The goal of this article is to study the structure and representation theory of affine wreath product algebras $\mathbf{W}{\mathbf{r}}_n^{\text{aff}}(A)$ and its cyclotomic quotients $\mathbf{W}{\mathbf{r}}_n^{\mathbf{C}}(A).$ These algebras appear naturally in Heisenberg categorification and generalize many important algebras (degenerate affine Hecke algebras, affine Sergeev algebras and wreath Hecke algebras). The whole class was introduced by D. Rosso and A. Savage in [16]. In [19], Savage studied both structure and representations under the condition that the trace map of A is even. In this paper we extend the definition for the case of odd trace. Since our approach is analogous to Savage’s, we consider the trace map being of arbitrary parity and unify statements and proofs. We also use an approach based on string diagrams, in the spirit of [5]. For simplicity of exposition, we assume A to be symmetric.

Keywords:

• Cyclotomic quotients
• Frobenius algebras
• Hecke algebras
• Mackey theorem
• odd wreath product algebras
• Sergeev algebras

2020 Mathematics subject classifications:

• Primary: 20C08
• Secondary: 18m30
• 18m05
• 17b10

Acknowledgement

I would like to thank Alistair Savage and Iryna Kashuba, for their guidance throughout the project and for helpful comments that improved the paper.

Additional information

Funding

This research was supported by FAPESP grant #2018/07628-9. Sao Paulo Research Foundation (FAPESP).