Polynomial functors and categorifications of Fock space

Hong, Jiuzu; Yacobi, Oded

doi:10.1007/978-1-4939-1590-3_12

Cited by 8 publications

(9 citation statements)

References 27 publications

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“…We hope this paper is a significant step in this program. This work is inspired by our previous works [HY1,HY2,HTY] on polynoimal functors and categorifications.…”

Section: Introductionmentioning

confidence: 99%

Quantum polynomial functors

Hong

Yacobi

2017

Journal of Algebra

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We construct a category of quantum polynomial functors which deforms Friedlander and Suslin's category of strict polynomial functors. The main aim of this paper is to develop from first principles the basic structural properties of this category (duality, projective generators, braiding etc.) in analogy with classical strict polynomial functors. We then apply the work of Hashimoto and Hayashi in this context to construct quantum Schur/Weyl functors, and use this to provide new and easy derivations of quantum (GL m , GL n ) duality, along with other results in quantum invariant theory.

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“…We hope this paper is a significant step in this program. This work is inspired by our previous works [HY1,HY2,HTY] on polynoimal functors and categorifications.…”

Section: Introductionmentioning

confidence: 99%

Quantum polynomial functors

Hong

Yacobi

2017

Journal of Algebra

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“…In view of the block structure of P p we restrict the poset P to the set of all hooks with a partial order induced by the partial order on P. By the Kuhn duality Ext q (W i , F j ) = Ext q (F j , S i ) for any 0 ≤ i, j ≤ p − 1 and q ≥ 0. Thus in the case of the category P p we can rewritethe equality (12) has the following form:…”

Section: Additive Ext Computationsmentioning

confidence: 99%

“…It was showed in [12] that if two blocks of P have the same p-weight then they are derived equivalent (c.f. Theorem 5, [12]).…”

Section: The Blocks Of P-weightmentioning

confidence: 99%

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On homological properties of strict polynomial functors of degree p

Jaśniewski¹

2021

Preprint

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We study the homological algebra in the category Pp of strict polynomial functors of degree p over a field of positive characteristic p. We determine the decomposition matrix of our category and we calculate the Ext-groups between functors important from the point of view of representation theory. Our results include computations of the Ext-algebras for simple functors and Schur functors and the Ext-groups between Schur and Weyl functors. We observe that the category Pp has a Kazhdan-Lusztig theory and we show that the dg algebras computing the Ext-algebras for simple functors and Schur functors are formal. These last results allow one to describe the bounded derived category of Pp as derived categories of certain explicitly described graded algebras. We also generalize our results to all blocks of p-weight 1 in Pe for e > p.

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“…Then, under the identification ̺, ·, · corresponds to (·, ·) (for details see [HTY,Propsition 6.4]). Let M ∈ P. Tensor product defines a functor T M : P → P given by T M (N ) = M ⊗ N .…”

Section: The Category Pmentioning

confidence: 99%

Polynomial functors and categorifications of Fock space II

Hong

Yacobi

2013

Advances in Mathematics

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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 affine Lie algebras. In the current work we continue the study of polynomial functors from the point of view of higher representation theory. Firstly, we categorify the Fock space representation of the Heisenberg algebra on the category of polynomial functors. Secondly, we construct commuting actions of the affine Lie algebra and the level p action of the Heisenberg algebra on the derived category of polynomial functors over a field of characteristic p > 0, thus weakly categorifying the Fock space representation of gl p . 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. The classical pictureWe begin by reviewing various structures related to the algebra of symmetric functions.

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Polynomial functors and categorifications of Fock space

Cited by 8 publications

References 27 publications

Quantum polynomial functors

Quantum polynomial functors

On homological properties of strict polynomial functors of degree p

Polynomial functors and categorifications of Fock space II

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