Schur-Weyl duality and Lie superalgebras
In the classical setting, Schur-Weyl duality describes an interaction between the symmetric group on d elements and the general linear Lie algebra gl(n), in terms of their action on d tensor copies of the vector representation of gl(n). This approach has been extended by Arakawa and Suzuki, and later Brundan and Kleshchev, to more general gl(n)-representations by upgrading the symmetric group to the degenerate affine Hecke algebra. A further generalization includes replacing gl(n) by sp(2n) or so(n), and the symmetric group by the Brauer algebra respectively. I will review some of these constructions and then discuss another instance of Schur-Weyl duality for the periplectic Lie superalgebra. One aspect which makes this case more unusual is the trivial action of the center of the universal enveloping algebra, and so a more elaborate construction than the standard Casimir element is required.
Iva Halacheva, University of Melbourne