Tensor diagrams and cluster algebras

Fomin, Sergey; Pylyavskyy, Pavlo

doi:10.1016/j.aim.2016.03.030

Cited by 46 publications

(113 citation statements)

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“…The following statement is a direct corollary of the natural extension of the Starfish Lemma [, Proposition 3.6] to the case of generalized cluster structures. The proof of this extension literally follows the proof of the Starfish Lemma.…”

Section: Preliminariesmentioning

confidence: 98%

Drinfeld double of GLn and generalized cluster structures

Gekhtman

Shapiro

Vainshtein

2017

Proc. London Math. Soc.

130

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Section: Preliminariesmentioning

confidence: 98%

Drinfeld double of GLn and generalized cluster structures

Gekhtman

Shapiro

Vainshtein

2017

Proc. London Math. Soc.

130

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“…The original conclusion in [12,Corollary 3.7] is that R ⊇ C(∆, x). However, the proof implies this stronger result (see the comment before the proof of [12, Proposition 3.6]).…”

Section: Upper Cluster Algebras An Amazing Property Of Cluster Algebmentioning

confidence: 99%

“…Cluster Algebras. We follow mostly Section 3 of [12]. The combinatorial data defining a cluster algebra is encoded in an ice quiver ∆ with no loops or oriented 2-cycles.…”

Section: Graded Cluster Algebrasmentioning

confidence: 99%

Cluster algebras, invariant theory, and Kronecker coefficients I

Fei

2017

Advances in Mathematics

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Abstract. We relate the m-truncated Kronecker products of symmetric functions to the semi-invariant rings of a family of quiver representations. We find cluster algebra structures for these semi-invariant rings when m = 2. Each g-vector cone G l of these cluster algebras controls the 2-truncated Kronecker products for all symmetric functions of degree no greater than l. As a consequence, each relevant Kronecker coefficient is the difference of the number of the lattice points inside two rational polytopes. We also give explicit description of all G l 's. As an application, we compute some invariant rings. IntroductionGiven a partition λ of n, let S λ be the associated irreducible complex representation of the symmetric group S n . The Kronecker coefficients g λ µ,ν are the tensor product multiplicities:To determine these coefficients and understand their properties has been one of the major problems in combinatorics and representation theory for nearly a century. People are particularly interested in finding combinatorial interpretation for these coefficients. They hope that Kronecker coefficients count some combinatorial objects, eg., lattice points in polytopes. More recently, the interest in computing Kronecker coefficients has intensified in connection with Geometric Complexity Theory [23], pioneered as an approach to the P vs. NP problem. Despite of a large body of work, those coefficients remain very mysterious. The complete solution of analogous problems for the general linear groups GL n certainly bring new hope to this old problem. The tensor multiplicities of irreducible representations of GL n , known as Littlewood-Richardson coefficients, can be computed by the Littlewood-Richardson rule. According to , they are counted by the lattice points in the so-called hive polytopes. Similar results were also obtained by 4] via an invariant-theoretic approach. The link between these two approachs was established in [10] through Fomin-Zelevinsky's cluster algebras [13,1,14] and their quiver with potential models [5,6]. In this paper, we use the similar ideas in [3,10] to study the Kronecker coefficients.2010 Mathematics Subject Classification. Primary 20C30, 13F60; Secondary 16G20, 13A50, 52B20.

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“…The positive rational expressions from the previous section are reminiscent of a cluster algebra structure (see [FP,§3] for definitions). In fact, (5.1.3) and (5.1.4) are exactly the exchange relations for the local moves in double wiring diagrams [FZP,Figure 9].…”

Section: N and Lm Nmentioning

confidence: 99%

“…We may directly apply the analogue of [FP,Proposition 3.6] for LP algebras (for which the proof is identical) to LM n as defined in Definition 6.2.2. It is well-known that minors of a matrix of indeterminates are irreducible, so we immediately have that all of our seed variables are pairwise coprime.…”

Section: N and Lm Nmentioning

confidence: 99%

Circular planar electrical networks: Posets and positivity

Alman

Lian

Tran

2015

Journal of Combinatorial Theory, Series A

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Following de Verdière-Gitler-Vertigan and Curtis-Ingerman-Morrow, we prove a host of new results on circular planar electrical networks. We first construct a poset EP n of electrical networks with n boundary vertices, and prove that it is graded by number of edges of critical representatives. We then answer various enumerative questions related to EP n , adapting methods of Callan and Stein-Everett. Finally, we study certain positivity phenomena of the response matrices arising from circular planar electrical networks. In doing so, we introduce electrical positroids, extending work of Postnikov, and discuss a naturally arising example of a Laurent phenomenon algebra, as studied by Lam-Pylyavskyy.

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Tensor diagrams and cluster algebras

Cited by 46 publications

References 43 publications

Drinfeld double of GLn and generalized cluster structures

Drinfeld double of GLn and generalized cluster structures

Cluster algebras, invariant theory, and Kronecker coefficients I

Circular planar electrical networks: Posets and positivity

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