Affine wreath product algebras with trace maps of generic parity

Author(s)

Eduardo Monteiro Mendonça

• Document type: Master's Dissertation
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI)
• Defense date: 2020-07-10

Examining board members

Iryna Kashuba; Lucas Henrique Calixto; Adriano Adrega de Moura

• Advisor: Iryna Kashuba; Alistair Savage

Abstract

The goal of this project is to study the structure and representation theory of affine wreath product algebras An(F). 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 [RS17]. In [Sav20] the second author studied both structure and representations under the condition that the trace map of F is even. In this project we extend the definition for the case where the trace is odd, obtaining ``odd-analogous\'\' results. Since our approach is analogous to Savage\'s, we consider the trace map being of arbitrary parity and unify statements and proofs. Studing the strucure theory, we present a basis for An(F) and compute its center. We also introduce Jucys-Murphy and intertwining elements. Considering a equivalence of categories, we describe the simple An(F)-modules in terms of simple representations of degenerate affine Hecke algebras, affine Sergeev algebras and wreath product algebras. We define the cyclotomic quotients AnC(F) for An(F) and we show that these are Frobenius algebras with an appropriately chosen trace map. We also state a cyclotomic Mackey Theorem and show that AnC(F) is a Frobenius extension of An-1C(F)$. (AU)

• FAPESP's process: 18/07628-9 - Affine wreath product algebras
• Grantee: Eduardo Monteiro Mendonça
• Support Opportunities: Scholarships in Brazil - Master