Promotion and cyclic sieving via webs

Petersen, T. Kyle; Pylyavskyy, Pavlo; Rhoades, Brendon

doi:10.1007/s10801-008-0150-3

Cited by 61 publications

(77 citation statements)

References 14 publications

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“…(We use the streamlined presentation of Petersen, Pylyavskyy, and Rhoades [8].) Boundary vertices correspond to copies of either the fundamental representation V of sl 3 or its dual V * , depending on whether the vertex is a source or a sink.…”

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confidence: 99%

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

Tymoczko

2011

J Algebr Comb

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“…7 of the survey [25] on the cyclic sieving phenomenon. The paper [23] gives a simpler proof of this result in the cases n = 2 and n = 3 using diagrams. The paper [30] gives a simpler proof of this result for all n using Lusztig's dual canonical basis.…”

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confidence: 88%

“…The proof in [30] can be made almost elementary by replacing the dual canonical basis with the basis constructed below. This essentially extends the method in [23] to all n.…”

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confidence: 90%

Web bases for the general linear groups

Westbury

2011

J Algebr Comb

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“…In the cases = 2, 3, there are proofs that proceed by giving a bijection to other combinatorial objects (noncrossing matchings and webs, respectively) that sends promotion to rotation [PePyRh09]; see Figure 7. There are also homomesy results in this and more general settings; see [BlPeSa16].…”

Section: Theorem 15 ([Rh10]) Promotion On ( × ) Exhibits the Cyclic mentioning

confidence: 99%

Dynamical Algebraic Combinatorics: Promotion, Rowmotion, and Resonance

Striker¹

2017

Notices Amer. Math. Soc.

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Mathematical inquiry often begins with the study of objects (numbers, shapes, variables, matrices, ideals, metric spaces, …) and the question, "What are the objects like?" It then moves to the study of actions (functions, rotations, reflections, multiplication, derivatives, shifts, …) and the question, "How do the objects behave?" The study of actions in various mathematical contexts has been extremely fruitful; consider the study of metric spaces through the lens of dynamical systems or the study of symmetries arising from group actions. For objects and actions arising from algebraic combinatorics, we call this study dynamical algebraic combinatorics.Let be a bijective action on a finite set . Such an action breaks the space into orbits. Often, the study of the orbits of provides insight into the structure of the objects in , revealing hidden symmetries and connections. One typically first seeks to understand the order of the action and then finds interesting properties the action exhibits. One surprisingly ubiquitous property is the cyclic sieving phenomenon [ReStWh04], which occurs when the evaluation of a generating function for at the primitive th root of unity (where = 2 / ) counts the number of elements of fixed under .For example, let be the set of binary words composed of two zeros and two ones. Let be the cyclic shift that acts by moving the first digit to the end of the word. Each word has four digits, so is of order four. Consider the inversion number statistic inv( ) of ∈ , which equals the number of pairs ( , ) with < such that the th digit of is 1 and the th digit is 0. See Figure 1 for the orbits of under and the inversion numbers of each binary word.

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Promotion and cyclic sieving via webs

Abstract: We show that Schützenberger's promotion on two and three row rectangular Young tableaux can be realized as cyclic rotation of certain planar graphs introduced by Kuperberg. Moreover, following work of the third author, we show that this action admits the cyclic sieving phenomenon.

Cited by 61 publications

References 14 publications

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

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

Web bases for the general linear groups

Dynamical Algebraic Combinatorics: Promotion, Rowmotion, and Resonance

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