Schur-Weyl duality and its variations

Yung-Ning Peng

Department of Mathematics

National Central University

ynp@math.ncu.edu.tw

    Let `V=\mathbb{C}^n` be the standard representation of the general linear Lie algebra `\mathcal{G}=\mathfrak{gl}(\mathbb{C})`. Then for any positive integer `d`, the tensor space `\mathbb{V}=V^{\otimes d}` is naturally a `\mathcal{G}`-module and hence a `U(\mathcal{G})` is the universal enveloping algebra of `\mathcal{G}`. On the other hand, the symmetric group `S_d` also acts naturally on `\mathbb{V}` by permuting the tensor factors. As a result, `\mathbb{V}`, can be viewed as a `\mathbb{C}[S_d]`-module as well, where `\mathbb{C}[S_d]` means the group algebra of `S_d`. The celebrated Schur-Weyl duality implies that the images of these two actions in the endomorphism space `\text{End}\mathbb{V}` actually form a full centralizer of each other.
In this short talk, we will try to introduce a few variations of the Schur-Weyl duality by replacing the role of `\mathcal{G}` by other structure (in particular, the periplectic Lie superalgebra `\mathfrak{p}_n`) and suitably modifying `\mathbb{V}`. In particular, an interesting algebra `A_d` and it's affine version, called the affine pereplectic Brauer algebra `"\hat{P}_d^-`, will show up. If time permitted, we will give a diagrammatic realization of `\hat{P}_d^-` together with a PBW basis which is very similar to that of the Brauer algebras and that of the Nazarov-Wenzl algebras. This talk is based on a joint work with Prof. Chih-Whi Chen (Xiamen University).

Keyword: Schur-Weyl duality, Brauer algebra, pereplectice Lie superalgebra.