Representation theory of the symmetric group can be 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 three main objectives.
1) We will classify the blocks of the periplectic Brauer category associated to the periplectic Brauer algebra. Ideally, we would like to have the decomposition numbers of the periplectic Brauer algebra. However, since this is a long-standing unsolved problem even for the symmetric group,
we aim to obtain a rougher description of the links between simple finite dimensional representations of the periplectic Brauer algebra. This brings our understanding to the same level as for the Brauer algebra.
2) We also aim to construct a categorical representation of the affine Temperley-Lieb algebra on a category associated with the periplectic Brauer algebra.
3) Finally, we want to find appropriate quantum deformations of the periplectic Brauer algebra.
They should satisfy a kind of Schur-Weyl duality with the quantum group of the periplectic Lie superalgebra. We will also initiate the study of their representation theory.