Springer representations on the Khovanov Springer varieties

Russell, Heather M.; Tymoczko, Julianna

doi:10.1017/s0305004111000132

Cited by 19 publications

(27 citation statements)

References 17 publications

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“…The proof of simple-connectivity is essentially the same as Seidel-Smith's proof that H 1 (Y n ) = 0 [53,Lemma 44]: as explained there (see also [53,Section 2.3]), Y n is diffeomorphic to the (n, n) Springer variety B n,n . Russell and Tymoczko [48,Theorem 5.5] and Wehrli [62,Theorem 1.2] showed that B n,n is homeomorphic to the space S := a S a ⊂ (CP 1 ) 2n where the union is over all isotopy classes a of upper half-plane crossingless matchings a, viewed as involutions a : {1, . .…”

Section: A Brief Review Of Symplectic Khovanov Homologymentioning

confidence: 99%

A flexible construction of equivariant Floer homology and applications

Hendricks

Lipshitz²,

Sarkar³

2016

J Topology

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Seidel–Smith and Hendricks used equivariant Floer cohomology to define some spectral sequences from symplectic Khovanov homology and Heegaard Floer homology. These spectral sequences give rise to Smith‐type inequalities. Similar‐looking spectral sequences have been defined by Lee, Bar–Natan, Ozsváth–Szabó, Lipshitz–Treumann, Szabó, Sarkar–Seed–Szabó, and others. In this paper, we give another construction of equivariant Floer cohomology with respect to a finite group action and use it to prove some invariance properties of these spectral sequences; prove that some of these spectral sequences agree; improve Hendricks's Smith‐type inequalities; give some theoretical and practical computability results for these spectral sequences; define some new spectral sequences conjecturally related to Sarkar–Seed–Szabó's; and introduce a new concordance homomorphism and concordance invariants. We also digress to prove invariance of Manolescu's reduced symplectic Khovanov homology.

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Section: A Brief Review Of Symplectic Khovanov Homologymentioning

confidence: 99%

A flexible construction of equivariant Floer homology and applications

Hendricks

Lipshitz²,

Sarkar³

2016

J Topology

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“…Khovanov and Kuperberg constructed a bijection between Young tableaux of shape (n, n, n) and irreducible webs for sl 3 for which each boundary vertex is a source [4]. However, unlike our direct map, Khovanov-Kuperberg's proof uses a complicated set of growth rules which, when recursively applied, eventually generate all irreducible webs for sl 3 . Our work is motivated by the geometry of the (n, n, n) Springer variety and generalizes earlier work on the (n, n) Springer variety of Fung [2] and of the author with Russell [10]. In that context, a web's boundary vertices correspond to the nested vector spaces of an element in the full flag variety, and edges correspond to lines in the vector space C 2n .…”

mentioning

confidence: 97%

A simple bijection between standard 3×n tableaux and irreducible webs for $\mathfrak{sl}_{3}$

Tymoczko

2011

J Algebr Comb

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Combinatorial spiders are a model for the invariant space of the tensor product of representations. The basic objects, webs, are certain directed planar graphs with boundary; algebraic operations on representations correspond to graph-theoretic operations on webs. Kuperberg developed spiders for rank 2 Lie algebras and sl 2 . Building on a result of Kuperberg, Khovanov-Kuperberg found a recursive algorithm giving a bijection between standard Young tableaux of shape 3 × n and irreducible webs for sl 3 whose boundary vertices are all sources.In this paper, we give a simple and explicit map from standard Young tableaux of shape 3 × n to irreducible webs for sl 3 whose boundary vertices are all sources and show that it is the same as Khovanov-Kuperberg's map. Our construction generalizes to some webs with both sources and sinks on the boundary. Moreover, it allows us to extend the correspondence between webs and tableaux in two ways. First, we provide a short, geometric proof of Petersen-Pylyavskyy-Rhoades's recent result that rotation of webs corresponds to jeu-de-taquin promotion on 3 × n tableaux. Second, we define another natural operation on tableaux called a shuffle and show that it corresponds to the join of two webs. Our main tool is an intermediary object between tableaux and webs that we call an m-diagram. The construction of m-diagrams, like many of our results, applies to shapes of tableaux other than 3 × n.

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“…( Remark 5. In contrast to the related work [RT11] in type A we work with the entire homology and identify it as an induced representation. This approach has the following advantages:…”

Section: Theorem Cmentioning

confidence: 99%

“…Remark 11. The white dots in this article play exactly the same role as the black dots on the cup diagrams in the related work in type A, [RT11], [Rus11]. In order to distinguish them from the black markers (which do not appear in type A) we chose to color them white.…”

Section: A Diagrammatic Homology Basis and The Proof Of Theorem Dmentioning

confidence: 99%

Two-block Springer fibers of types C and D: a diagrammatic approach to Springer theory

Stroppel

Wilbert

2018

Math. Z.

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We explain an elementary topological construction of the Springer representation on the homology of (topological) Springer fibers of types C and D in the case of nilpotent endomorphisms with two Jordan blocks. The Weyl group and component group actions admit a diagrammatic description in terms of cup diagrams which appear in the definition of arc algebras of types B and D. We determine the decomposition of the representations into irreducibles and relate our construction to classical Springer theory. As an application we obtain presentations of the cohomology rings of all two-block Springer fibers of types C and D. Moreover, we deduce explicit isomorphisms between the Kazhdan-Lusztig cell modules attached to the induced trivial module and the irreducible Specht modules in types C and D.

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Springer representations on the Khovanov Springer varieties

Cited by 19 publications

References 17 publications

A flexible construction of equivariant Floer homology and applications

A flexible construction of equivariant Floer homology and applications

A simple bijection between standard 3×n tableaux and irreducible webs for $\mathfrak{sl}_{3}$

Two-block Springer fibers of types C and D: a diagrammatic approach to Springer theory

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