TY - JOUR
T1 - Affine Super Schur Duality To the memory of Goro Shimura
AU - Flicker, Yuval Z.
N1 - Publisher Copyright:
© 2023 Research Institute for Mathematical Sciences, Kyoto University.
PY - 2023
Y1 - 2023
N2 - 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 ↦→ M ⊗C[Sd ] E⊗d, E now being the super vector space Cm|n, from the category of finite-dimensional C[Sd ⋉ ℤd ]-modules, or representations of the affine Weyl, or symmetric, group Sda = Sd ⋉ ℤd, to the category of finite-dimensional representations of the universal enveloping algebra of the affine Lie superalgebraU(ŝl(m|n)) that are E⊗dcompatible, 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 ŝl(m|n) are tensor products of evaluation representations at distinct points of C×.
AB - 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 ↦→ M ⊗C[Sd ] E⊗d, E now being the super vector space Cm|n, from the category of finite-dimensional C[Sd ⋉ ℤd ]-modules, or representations of the affine Weyl, or symmetric, group Sda = Sd ⋉ ℤd, to the category of finite-dimensional representations of the universal enveloping algebra of the affine Lie superalgebraU(ŝl(m|n)) that are E⊗dcompatible, 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 ŝl(m|n) are tensor products of evaluation representations at distinct points of C×.
KW - Affine symmetric group
KW - affine Lie superalgebra
KW - affine Schur duality
KW - ŝl(m|n)
UR - http://www.scopus.com/inward/record.url?scp=85149935683&partnerID=8YFLogxK
U2 - 10.4171/PRIMS/59-1-5
DO - 10.4171/PRIMS/59-1-5
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AN - SCOPUS:85149935683
SN - 0034-5318
VL - 59
SP - 153
EP - 202
JO - Publications of the Research Institute for Mathematical Sciences
JF - Publications of the Research Institute for Mathematical Sciences
IS - 1
ER -