Polynomial functors and categorifications of Fock space II: Schur-Weyl duality

Details Polynomial functors and categorifications of Fock space II: Schur-Weyl duality Polynomial functors and categorifications of Fock space II: Schur-Weyl duality

2011-11-22

arxiv.org/abs/1111.5335v1

Mathematics

Representation Theory

44 pages

Scientific paper

We categorify various Fock space representations on the algebra of symmetric functions via the category of polynomial functors. In a prequel, we used polynomial functors to categorify the Fock space representations of type A Kac-Moody algebras. In the current work we define a representation of the Heisenberg algebra on the category of polynomial functors, and show that it categorifies the Fock space representation. We also categorify the commuting actions of the affine Lie algebras and the Heisenberg algebras on Fock space. Moreover, we study the relationship between these categorifications and Schur-Weyl duality. The duality is formulated as a functor from the category of polynomial functors to the category of linear species. The category of linear species is known to carry actions of the Kac-Moody algebra and the Heisenberg algebra. We prove that Schur-Weyl duality is a morphism of these categorification structures.

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