Jucys–Murphy elements, orthogonal matrix integrals, and Jack measures

Matsumoto, Sho

doi:10.1007/s11139-011-9317-y

Cited by 27 publications

(46 citation statements)

References 21 publications

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“…In particular, one has |µ| = |π| − ℓ(π) and z µ+(1 n−|µ| ) = z π . Besides, F (C λ ) in our paper corresponds to α d/2 F (A α λ ) in [Mat10]. Finally, the probability P (α) n (λ) is simply given by n!α n J λ ,J λ .…”

Section: Plugging This Into Equationmentioning

confidence: 88%

Gaussian fluctuations of Young diagrams and structure constants of Jack characters

Dołęga

Féray

2016

Duke Math. J.

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In this paper, we consider a deformation of Plancherel measure linked to Jack polynomials. Our main result is the description of the first and second-order asymptotics of the bulk of a random Young diagram under this distribution, which extends celebrated results of Vershik-Kerov and Logan-Shepp (for the first order asymptotics) and Kerov (for the second order asymptotics). This gives more evidence of the connection with Gaussian β-ensemble, already suggested by some work of Matsumoto.Our main tool is a polynomiality result for the structure constant of some quantities that we call Jack characters, recently introduced by Lassalle. We believe that this result is also interested in itself and we give several other applications of it.2010 Mathematics Subject Classification. 60C05, 05E05 (secondary:60B20).

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Section: Plugging This Into Equationmentioning

confidence: 88%

Gaussian fluctuations of Young diagrams and structure constants of Jack characters

Dołęga

Féray

2016

Duke Math. J.

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“…Let us note that this revisited approach to Weingarten calculus is related to, and implies results of [MN13] in the unitary case and from [ZJ10,M11] in the orthogonal case.…”

Section: Introductionmentioning

confidence: 85%

Weingarten calculus via orthogonality relations: new applications

Collins¹,

Matsumoto²

2017

ALEA

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Weingarten calculus is a completely general and explicit method to compute the moments of the Haar measure on compact subgroups of matrix algebras. Particular cases of this calculus were initiated by theoretical physicists -including Weingarten, after whom this calculus was coined by the first author, after investigating it systematically. Substantial progress was achieved subsequently by the second author and coworkers, based on representation theoretic and combinatorial techniques. All formulas of 'Weingarten calculus' are in the spirit of Weingarten's seminal paper [W78]. However, modern proofs are very different from Weingarten's initial ideas. In this paper, we revisit Weingarten's initial proof and we illustrate its power by uncovering two new important applications: (i) a uniform bound on the Weingarten function, that subsumes existing uniform bounds, and is optimal up to a polynomial factor, and (ii) an extension of Weingarten calculus to symmetric spaces and conceptual proofs of identities established by the second author.

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“…h(a) = h( a) = a for all a ∈ [n]. Matsumoto defined the following analogue of monotone factorizations [31]. Let m be a matching and let (τ 1 , ..., τ k ) be a sequence of transpositions τ i = (s i t i ), in which all t i ∈ [n] with t i ≥ t i−1 and t i > h(s i ), such that m = τ 1 · · · τ k (t), where t is the trivial matching.…”

Section: Orthogonal Casementioning

confidence: 99%

Expansion of polynomial Lie group integrals in terms of certain maps on surfaces, and factorizations of permutations

Novaes¹

2017

J. Phys. A: Math. Theor.

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Using the diagrammatic approach to integrals over Gaussian random matrices, we find a representation for polynomial Lie group integrals as infinite sums over certain maps on surfaces. The maps involved satisfy a specific condition: they have some marked vertices, and no closed walks that avoid these vertices. We also formulate our results in terms of permutations, arriving at new kinds of factorization problems.

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Jucys–Murphy elements, orthogonal matrix integrals, and Jack measures

Cited by 27 publications

References 21 publications

Gaussian fluctuations of Young diagrams and structure constants of Jack characters

Gaussian fluctuations of Young diagrams and structure constants of Jack characters

Weingarten calculus via orthogonality relations: new applications

Expansion of polynomial Lie group integrals in terms of certain maps on surfaces, and factorizations of permutations

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