We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
Affine Super Schur Duality: To the memory of Goro Shimura.
- Authors
FLICKER, Yuval Z.
- Abstract
Schur duality is an equivalence, for d ≤ n, between the category of finite-dimensional representations over C of the symmetric group Sd on d letters, and the category of finite-dimensional representations over C of GL(n,C) whose irreducible subquotients are subquotients of E⊗d, E = Cn. The latter are called polynomial representations homogeneous of degree d. It is based on decomposing E⊗d as a C[Sd] × GL(n,C)-bimodule. It was used by Schur to conclude the semisimplicity of the category of finite-dimensional complex GL(n,C)-modules from the corresponding result for Sd that had been obtained by Young. Here we extend this duality to the affine super case by constructing a functor F: M 7→ M ⊗C[Sd] E⊗d, E now being the super vector space Cm|n, from the category of finite-dimensional C[Sd x Zd]-modules, or representations of the affine Weyl, or symmetric, group Sa d = Sd x Zd, to the category of finite-dimensional representations of the universal enveloping algebra of the affine Lie superalgebra U(bsl(m|n)) that are E⊗d-compatible, namely the subquotients of whose restriction to U(sl(m|n)) are constituents of E⊗d. Both categories are not semisimple. When d < m+n the functor defines an equivalence of categories. As an application we conclude that the irreducible finite-dimensional E⊗d-compatible representations of the affine superalgebra bsl(m|n) are tensor products of evaluation representations at distinct points of C×.
- Subjects
SEMISIMPLE Lie groups; LIE superalgebras; UNIVERSAL algebra; AFFINE algebraic groups; HOMOGENEOUS polynomials; TENSOR products; VECTOR spaces; LIE algebras
- Publication
Publications of the Research Institute for Mathematical Sciences, 2023, Vol 59, Issue 1, p153
- ISSN
0034-5318
- Publication type
Article
- DOI
10.4171/PRIMS/59-1-5