Polynomial functors and categorifications of Fock space II: Schur-Weyl duality
Polynomial functors and categorifications of Fock space II: Schur-Weyl duality
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Fock 空间 II 的多项式函子和分类：Schur-Weyl 对偶性
DOI：
10.1007/978-1-4939-1590-3_12
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发表时间：
2011
期刊：
影响因子：
--
通讯作者：
Oded Yacobi
中科院分区：
文献类型：
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作者：
Jiuzu Hong;Oded Yacobi
我们通过多项式函子范畴对对称函数代数上的各种Fock空间表示进行分类。在前传中，我们使用多项式函子来分类A型Kac-Moody代数的Fock空间表示。在本文中，我们定义了Heisenberg代数在多项式函子范畴上的一种表示，并证明了它是Fock空间表示的范畴。并对仿射李代数和Heisenberg代数在Fock空间上的交换作用进行了分类。此外，我们还研究了这些范畴与Schur-Weyl对偶之间的关系。将对偶性表示为从多项式函子范畴到线性物种范畴的函子。众所周知，线性物种范畴具有Kac-Moody代数和Heisenberg代数的作用。我们证明了Schur-Weyl对偶是这些范畴结构的态射。
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.