Three perspectives on categorical symmetric Howe duality

Webster, Ben

doi:10.48550/arxiv.2001.07584

Cited by 4 publications

(5 citation statements)

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“…Strands of different types stand out in the construction of Webster algebras and associated link homology theories [We1]. Relation in Figure 3.0.8 with a minor variation appears in the redotted Webster algebra case, see [KLSY,Section 4.2] and [We2], and an approach to these algebras, bimodules and associated link invariants via multi-type overlapping foam evaluations may exist as well.…”

Section: Overlapping Theta-foams and The Sergeev-pragacz Formulamentioning

confidence: 99%

Universal construction of topological theories in two dimensions

Khovanov

2020

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We consider Blanchet, Habegger, Masbaum and Vogel's universal construction of topological theories in dimension two, using it to produce interesting theories that do not satisfy the usual two-dimensional TQFT axioms. Kronecker's characterization of rational functions allows us to classify theories over a field with finite-dimensional state spaces and introduce their extension to theories with the ground ring the product of rings of symmetric functions in N and M variables. We look at several examples of non-multiplicative theories and see Hankel matrices, Schur and supersymmetric Schur polynomials quickly emerge from these structures. The last section explains how an extension of the Robert-Wagner foam evaluation to overlapping foams gives the Sergeev-Pragacz formula for the supersymmetric Schur polynomials and the Day formula for the Toeplitz determinant of rational power series as special cases. Contents 1. Universal construction in n dimensions 1 2. Universal construction in two dimensions 6 2.1. State space of a circle and the generating function 6 2.2. State space and the Hankel matrix 9 2.3. Rational theories over a field. 12 2.4. Semi-universal rational theories α M,N 13 2.5. One-circle state space for a polynomial generating function. 19 2.6. State spaces for unions of circles 22 2.7. Example of the constant generating function 26 2.8. Free theory 31 2.9. More examples 32 2.10. Recursive sequences 38 3. Overlapping theta-foams and the Sergeev-Pragacz formula 40 References 53 1. Universal construction in n dimensionsConsider the tensor category Cob n of oriented n-dimensional cobordisms. Its objects are oriented closed (n − 1)-manifolds N . Morphisms from N 0 to N 1 are equivalence classes of oriented compact n-manifolds M with ∂M = (−N 0 ) N 1 , and the equivalence relation is diffeomorphism rel boundary. This is a symmetric tensor category, and by a tensor category in this note we will mean a symmetric tensor category.

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Section: Overlapping Theta-foams and The Sergeev-pragacz Formulamentioning

confidence: 99%

Universal construction of topological theories in two dimensions

Khovanov

2020

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“…This fact is shown based on the theory of KLR algebras in [Weba, Thm 5.9]: we know a complete classification of graded simple T, and can show that none are killed by all idempotents of the form e(γ) for γ ∈ MaxSpec(Γ) Z,χ . However, as noted in the proof of [Webd,Cor. 4.10], a little work on the side of representation theory allows us to avoid using this fact for…”

Section: In Essence X±mentioning

confidence: 77%

“…The functor T sends the Verma module M (σ(µ − ρ) + ρ) to the Verma module M (σ(µ − ρ) + ρ); note that the singularity of the block means that we will sometimes have M (σ) ∼ = M (σ ) for two different permutations. On the other hand, by [Webd,Cor. 4.14], we have that the functor GT (which depends on a choice of central character) commutes with translation onto a wall.…”

Section: The Non-integral Casementioning

confidence: 93%

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Gelfand-Tsetlin modules: canonicity and calculations

Silverthorne

Webster

2020

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In this paper, we give a more down-to-earth introduction to the connection between Gelfand-Tsetlin modules over gl n and diagrammatic KLRW algebras, and develop some of its consequences. In addition to a new proof of this description of the category Gelfand-Tsetlin modules appearing in earlier work, we show three new results of independent interest: (1) we show that every simple Gelfand-Tsetlin module is a canonical module in the sense of Early, Mazorchuk and Vishnyakova, and characterize when two maximal ideals have isomorphic canonical modules, (2) we show that the dimensions of Gelfand-Tsetlin weight spaces in simple modules can be computed using an appropriate modification of Leclerc's algorithm for computing dual canonical bases, and (3) we construct a basis of the Verma modules of sl n which consists of generalized eigenvectors for the Gelfand-Tsetlin subalgebra.Furthermore, we present computations of multiplicities and Gelfand-Kirillov dimensions for all integral Gelfand-Tsetlin modules in ranks 3 and 4; unfortunately, for ranks > 4, our computers are not adequate to perform these computations.

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“…We also constructed the braid group action on the homotopy category of W (s, k). Subsequently, in the type A n , Webster studied a generalization for W (s, k) associated with Gelfand-Tsetlin modules [Web20].…”

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confidence: 99%