Stability Patterns in Representation Theory

Sam, Steven V; Snowden, Andrew

doi:10.1017/fms.2015.10

Cited by 70 publications

(60 citation statements)

References 43 publications

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“…In particular, these coefficients have been linked to Geometric Complexity Theory (see [2,12]), stable representation theory and FI-modules (see [5,15]) and to the study of the Fourier-Deligne transform (see [9]). …”

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confidence: 99%

Deligne categories and reduced Kronecker coefficients

Entova-Aizenbud

2016

J Algebr Comb

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The Kronecker coefficients are the structural constants for the tensor categories of representations of the symmetric groups, namely, given three partitions λ, μ, τ of n, the multiplicity of λ in μ ⊗ τ is called the Kronecker coefficient g λ μ,τ . When the first part of each of the partitions is taken to be very large (the remaining parts being fixed), the values of the appropriate Kronecker coefficients stabilize; the stable value is called the reduced (or stable) Kronecker coefficient. These coefficients also generalize the Littlewood-Richardson coefficients and have been studied quite extensively. In this paper, we show that reduced Kronecker coefficients appear naturally as structure constants of Deligne categories Rep(S t ). This allows us to interpret various properties of the reduced Kronecker coefficients as categorical properties of Deligne categories Rep(S t ) and derive new combinatorial identities.

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confidence: 99%

Deligne categories and reduced Kronecker coefficients

Entova-Aizenbud

2016

J Algebr Comb

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“…Sections 2 and 3 contain preliminaries on the Deligne category Rep(S ν ) (thoughout the paper, we use the parameter ν instead of t), the categories of polynomial representations of gl N (N ∈ Z + ∪ {∞}) and the parabolic category O for gl N . These sections are based on [5], [6] and [12].…”

Section: 2mentioning

confidence: 99%

“…The representations of the Lie algebra gl ∞ are discussed in detail in [11], [3], as well as [12,Section 3].…”

Section: Deligne Category Rep(s ν )mentioning

confidence: 99%

“…Note that Rep(gl ∞ ) poly is the free abelian symmetric monoidal category generated by one object (see [12, (2.2.11)]). It has a equivalent definition as the category of polynomial functors of bounded degree, which can be found in [9] and in [12].…”

Section: 2mentioning

confidence: 99%

“…The following Lemma is proved in [11], [12,Section 3]: Lemma 3.3.2. The functors Γ n are additive symmetric monoidal functors between semisimple symmetric monoidal categories.…”

Section: 2mentioning

confidence: 99%

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Schur–Weyl duality for Deligne categories, II : The limit case

Entova-Aizenbud

2016

Pacific J. Math.

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Abstract. This paper is a continuation of a previous paper of the author ([5]), which gave an analogue to the classical Schur-Weyl duality in the setting of Deligne categories.Given a finite-dimensional unital vector space V (a vector space V with a chosen non-zero vector ½), we constructed in [5] a complex tensor power of V : an Ind-object of the Deligne category Rep(St) which is a Harish-Chandra module for the pair (gl(V ),P½), whereP½ ⊂ GL(V ) is the mirabolic subgroup preserving the vector ½.This construction allowed us to obtain an exact contravariant functor SW t,V from the category Rep ab (St) (the abelian envelope of the category Rep(St)) to a certain localization of the parabolic category O associated with the pair (gl(V ),P½).In this paper, we consider the case when V = C ∞ . We define the appropriate version of the parabolic category O and its localization, and show that the latter is equivalent to a "restricted" inverse limit of categories O p t,C N with N tending to infinity. The Schur-Weyl functors SW t,C N then give an anti-equivalence between this category and the category Rep ab (St).This duality provides an unexpected tensor structure on the category O p∞ t,C ∞ .

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Three Results on Representations of Mackey Lie Algebras

Chirvăsitu

2014

Developments in Mathematics

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I. Penkov and V. Serganova have recently introduced, for any non-degenerate pairing W ⊗ V → C of vector spaces, the Lie algebra gl M = gl M (V, W ) consisting of endomorphisms of V whose duals preserve W ⊆ V * . In their work, the category T gl M of gl M -modules which are finite length subquotients of the tensor algebra T (W ⊗ V ) is singled out and studied. In this note we solve three problems posed by these authors concerning the categories T gl M . Denoting by T V ⊗W the category with the same objects as T gl M but regarded as V ⊗ W -modules, we first show that when W and V are paired by dual bases, the functor T gl M → T V ⊗W taking a module to its largest weight submodule with respect to a sufficiently nice Cartan subalgebra of V ⊗ W is a tensor equivalence. Secondly, we prove that when W and V are countable-dimensional, the objects of T End(V ) have finite length as gl M -modules. Finally, under the same hypotheses, we compute the socle filtration of a simple object in T End(V ) as a gl M -module.

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Stability Patterns in Representation Theory

Cited by 70 publications

References 43 publications

Deligne categories and reduced Kronecker coefficients

Deligne categories and reduced Kronecker coefficients

Schur–Weyl duality for Deligne categories, II : The limit case

Three Results on Representations of Mackey Lie Algebras

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