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

Tymoczko, Julianna

doi:10.1007/s10801-011-0317-1

Cited by 25 publications

(36 citation statements)

References 8 publications

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“…Kuperberg [42] showed that web invariants are a basis for C[Gr(3, m)], and with Khovanov [41] gave a bijective labeling T → [W (T )] of these webs by tableaux T ∈ SSYT (3, [m]). For an easily computable description of this bijection we refer to [62]. One useful property of the bijection is the following: if T has an i weakly northeast of an i + 1, then boundary vertices i, i + 1 are joined by a "fork" or "Y" in W (T ) (see Example 6.4 for an illustration of the meaning of "fork").…”

Section: Fomin and Pylyavskyy's Conjecturesmentioning

confidence: 99%

“…The second line is its homogeneous lift ch(T ) ∈ C[Gr (3,6)] expressed in the basis of standard monomials. This element of C[Gr (3,6)] is a web invariant with diagram , and this web is labeled by T in the Khovanov-Kuperberg bijection (see e.g [62,Section 3.2]). This is one of the two non-Plücker cluster variables in C[Gr (3,6)] (it is Y 123456 in the notation of [58]).…”

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confidence: 99%

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Quantum affine algebras and Grassmannians

et al. 2020

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We study the relation between quantum affine algebras of type A and Grassmannian cluster algebras. Hernandez and Leclerc described an isomorphism from the Grothendieck ring of a certain subcategory C ℓ of U q ( sl n )-modules to a quotient of the Grassmannian cluster algebra in which certain frozen variables are set to 1. We explain how this induces an isomorphism between the monoid of dominant monomials, used to parameterize simple modules, and a quotient of the monoid of rectangular semistandard Young tableaux with n rows and with entries in [n + ℓ + 1]. Via the isomorphism, we define an element ch(T ) in a Grassmannian cluster algebra for every rectangular tableau T . By results of Kashiwara, Kim, Oh, and Park, and also of Qin, every Grassmannian cluster monomial is of the form ch(T ) for some T . Using a formula of Arakawa-Suzuki, we give an explicit expression for ch(T ), and also give explicit q-character formulas for finite-dimensional U q ( sl n )-modules. We give a tableau-theoretic rule for performing mutations in Grassmannian cluster algebras. We suggest how our formulas might be used to study reality and primeness of modules, and compatibility of cluster variables.

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Section: Fomin and Pylyavskyy's Conjecturesmentioning

confidence: 99%

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confidence: 99%

Quantum affine algebras and Grassmannians

et al. 2020

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“…Our second observation is as follows. When r = 2 or 3, there is a well-known bijection between standard young tableaux and non-elliptic webs, due to Khovanov and Kuperberg [5] (see also [18]). It has the property that rotation of webs is given by promotion of standard young tableaux, and this can be used to give an elementary proof [15] of the cyclic sieving phenomenon for rectangular tableaux with 2 or 3 or rows.…”

Section: Appendix: Web Duality Picturesmentioning

confidence: 99%

From dimers to webs

Fraser

Lam

2019

Trans. Amer. Math. Soc.

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We formulate a higher-rank version of the boundary measurement map for weighted planar bipartite networks in the disk. It sends a network to a linear combination of SL r -webs, and is built upon the r-fold dimer model on the network. When r equals 1, our map is a reformulation of Postnikov's boundary measurement used to coordinatize positroid strata. When r equals 2 or 3, it is a reformulation of the SL 2 -and SL 3 -web immanants defined by the second author. The basic result is that the higher rank map factors through Postnikov's map. As an application, we deduce generators and relations for the space of SL r -webs, reproving a result of Cautis-Kamnitzer-Morrison. We establish compatibility between our map and restriction to positroid strata, and thus between webs and total positivity.

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“…and resolving according to standard knot-theoretic relations. In this paper, as in much other combinatorial work [29,34], we focus on webs for (V + ) ⊗3n .…”

Section: Introductionmentioning

confidence: 99%

The Robinson–Schensted correspondence and $$A_2$$ A 2 -web bases

2015

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We study natural bases for two constructions of the irreducible representation of the symmetric group corresponding to [n, n, n]: the reduced web basis associated to Kuperberg's combinatorial description of the spider category; and the left cell basis for the left cell construction of Kazhdan and Lusztig. In the case of [n, n], the spider category is the Temperley-Lieb category; reduced webs correspond to planar matchings, which are equivalent to left cell bases. This paper compares the image of these bases under classical maps: the Robinson-Schensted algorithm between permutations and Young tableaux and Khovanov-Kuperberg's bijection between Young tableaux and reduced webs.One main result uses Vogan's generalized τ -invariant to uncover a close structural relationship between the web basis and the left cell basis. Intuitively, generalized τ -invariants refine the data of the inversion set of a permutation. We define generalized τ -invariants intrinsically for Kazhdan-Lusztig left cell basis elements and for webs. We then show that the generalized τ -invariant is preserved by these classical maps. Thus, our result allows one to interpret Khovanov-Kuperberg's bijection as an analogue of the Robinson-Schensted correspondence.Despite all of this, our second main result proves that the reduced web and left cell bases are inequivalent; that is, these bijections are not S 3n -equivariant maps.arXiv:1307.6487v2 [math.RT]

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

Cited by 25 publications

References 8 publications

Quantum affine algebras and Grassmannians

Quantum affine algebras and Grassmannians

From dimers to webs

The Robinson–Schensted correspondence and $$A_2$$ A 2 -web bases

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