Project

Modular representation theory of the periplectic Brauer algebra.

Duration
01 November 2020 → Ongoing
Funding
Research Foundation - Flanders (FWO)
Research disciplines
  • Natural sciences
    • Associative rings and algebras
    • Non-associative rings and algebras
    • Topological groups, Lie groups
    • Abstract harmonic analysis
Keywords
Representation theory Lie superalgebras finite-dimensional associative algebras.
 
Project description

Representation theory of the symmetric group is related to representation theory of the general linear group via Schur-Weyl duality. Similarly, Schur-Weyl duality also relates the orthogonal Lie group, the symplectic Lie group and the encompassing orthosymplectic Lie supergroup to the Brauer algebra, and it relates the periplectic Lie supergroup to the periplectic Brauer algebra. The goal of the project is to develop modular representation theory for the periplectic Brauer algebra. We have two main objectives and two smaller ones. 1) We aim to construct a categorical representation on the periplectic Deligne category associated to the periplectic Brauer algebra. We will then apply this to classify thick tensor ideals. 2) We want to find appropriate quantum deformations of the periplectic Brauer algebra. They should satisfy some Schur-Weyl duality with the quantum group of the periplectic Lie superalgebra. We will also initiate the study of their representation theory. 3) In the current state of the art, surjectivity for the periplectic Schur-Weyl duality in positive characteristic is only conjectured. We aim to prove this conjecture, raising our level of understanding to the same level as the other classical Lie superalgebras. 4) Finally we will apply our new knowledge on the modular representation theory of the periplectic Brauer algebra to obtain the blocks of the restricted modules of the periplectic Lie superalgebra in positive characteristic.