The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
Nicolai Reshetikhin - One of the best experts on this subject based on the ideXlab platform.
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on multiplicities of irreducibles in large tensor product of representations of simple lie algebras
Letters in Mathematical Physics, 2020Co-Authors: Olga Postnova, Nicolai ReshetikhinAbstract:In this paper, we study the asymptotics of multiplicities of irreducible representations in large tensor products of finite dimensional representations of simple Lie algebras and their statistics with respect to Plancherel and character probability measures. We derive the asymptotic distribution of irreducible components for the Plancherel measure, generalizing results of Biane and Tate and Zelditch. We also derive the asymptotics of the character measure for generic parameters and an intermediate scaling in the vicinity of the Plancherel measure. It is interesting that the asymptotic measure is universal and after suitable renormalization does not depend on which representations were multiplied but depends significantly on the degeneracy of the parameter in the character distribution.
Henrik Schlichtkrull - One of the best experts on this subject based on the ideXlab platform.
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The Plancherel decomposition for a reductive symmetric space. I. Spherical functions
Inventiones Mathematicae, 2005Co-Authors: Henrik SchlichtkrullAbstract:We prove the Plancherel formula for spherical Schwartz functions on a reductive symmetric space. Our starting point is an inversion formula for spherical smooth compactly supported functions. The latter formula was earlier obtained from the most continuous part of the Plancherel formula by means of a residue calculus. In the course of the present paper we also obtain new proofs of the uniform tempered estimates for normalized Eisenstein integrals and of the Maass–Selberg relations satisfied by the associated C-functions.
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the Plancherel decomposition for a reductive symmetric space ii representation theory
arXiv: Representation Theory, 2001Co-Authors: E P Van Den Ban, Henrik SchlichtkrullAbstract:We obtain the Plancherel decomposition for a reductive symmetric space in the sense of representation theory. Our starting point is the Plancherel formula for spherical Schwartz functions, obtained in part I (math.RT/0107063). The formula for Schwartz functions involves Eisenstein integrals obtained by a residual calculus. In the present paper we identify these integrals as matrix coefficients of the generalized principal series.
Yurii A Neretin - One of the best experts on this subject based on the ideXlab platform.
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Plancherel formula for berezin deformation of l2 on riemannian symmetric space
Journal of Functional Analysis, 2002Co-Authors: Yurii A NeretinAbstract:Consider natural representations of the pseudounitary group U(p, q) in the space of holomorphic functions on the Cartan domain (Hermitian symmetric space) U(p, q)/(U(p)×U(q)). Berezin representations of O(p, q) are the restrictions of such representations to the subgroup O(p, q). We obtain the explicit Plancherel formula for the Berezin representations. The support of the Plancherel measure is a union of many series of representations. The density of the Plancherel measure on each piece of the support is an explicit product of Γ-functions. We also show that the Berezin representations give an interpolation between L2 on noncompact symmetric space O(p, q)/O(p)×O(q) and L2 on compact symmetric space O(p+q)/O(p)×O(q).
N Shimeno - One of the best experts on this subject based on the ideXlab platform.
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the Plancherel formula for spherical functions with a one dimensional k type on a simply connected simple lie group of hermitian type
Journal of Functional Analysis, 1994Co-Authors: N ShimenoAbstract:Abstract We generalize the Harish-Chandra inversion formula for the spherical transform on a Riemannian symmetric space to homogeneous line bundles on a Hermitian symmetric space. We determine the Plancherel measure explicitly.
Grigori Olshanski - One of the best experts on this subject based on the ideXlab platform.
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Plancherel averages remarks on a paper by stanley
arXiv: Combinatorics, 2009Co-Authors: Grigori OlshanskiAbstract:Let M_n stand for the Plancherel measure on Y_n, the set of Young diagrams with n boxes. A recent result of Stanley (arXiv:0807.0383) says that for certain functions G defined on the set Y of all Young diagrams, the average of G with respect to M_n depends on n polynomially. We propose two other proofs of this result together with a generalization to the Jack deformation of the Plancherel measure.
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asymptotics of Plancherel type random partitions
Journal of Algebra, 2007Co-Authors: Alexei Borodin, Grigori OlshanskiAbstract:Abstract We present a solution to a problem suggested by Philippe Biane: we prove that a certain Plancherel-type probability distribution on partitions converges, as partitions get large, to a new determinantal random point process on the set Z + of nonnegative integers. This can be viewed as an edge limit transition. The limit process is determined by a correlation kernel on Z + which is expressed through the Hermite polynomials, we call it the discrete Hermite kernel. The proof is based on a simple argument which derives convergence of correlation kernels from convergence of unbounded self-adjoint difference operators. Our approach can also be applied to a number of other probabilistic models. As an example, we discuss a bulk limit for one more Plancherel-type model of random partitions.
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stochastic dynamics related to Plancherel measure on partitions
arXiv: Mathematical Physics, 2004Co-Authors: Alexei Borodin, Grigori OlshanskiAbstract:Consider the standard Poisson process in the first quadrant of the Euclidean plane, and for any point (u,v) of this quadrant take the Young diagram obtained by applying the Robinson-Schensted correspondence to the intersection of the Poisson point configuration with the rectangle with vertices (0,0), (u,0), (u,v), (0,v). It is known that the distribution of the random Young diagram thus obtained is the poissonized Plancherel measure with parameter uv. We show that for (u,v) moving along any southeast-directed curve in the quadrant, these Young diagrams form a Markov chain with continuous time. We also describe these chains in terms of jump rates. Our main result is the computation of the dynamical correlation functions of such Markov chains and their bulk and edge scaling limits.
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asymptotics of Plancherel measures for symmetric groups
Journal of the American Mathematical Society, 2000Co-Authors: Alexei Borodin, Andrei Okounkov, Grigori OlshanskiAbstract:1.1. Plancherel measures. Given a finite group G, by the corresponding Plancherel measure we mean the probability measure on the set G∧ of irreducible representations of G which assigns to a representation π ∈ G∧ the weight (dim π)/|G|. For the symmetric group S(n), the set S(n)∧ is the set of partitions λ of the number n, which we shall identify with Young diagrams with n squares throughout this paper. The Plancherel measure on partitions λ arises naturally in representation– theoretic, combinatorial, and probabilistic problems. For example, the Plancherel distribution of the first part of a partition coincides with the distribution of the longest increasing subsequence of a uniformly distributed random permutation [31]. We denote the Plancherel measure on partitions of n by Mn,
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asymptotics of Plancherel measures for symmetric groups
arXiv: Combinatorics, 1999Co-Authors: Alexei Borodin, Andrei Okounkov, Grigori OlshanskiAbstract:We consider the asymptotics of the Plancherel measures on partitions of $n$ as $n$ goes to infinity. We prove that the local structure of a Plancherel typical partition (which we identify with a Young diagram) in the middle of the limit shape converges to a determinantal point process with the discrete sine kernel. On the edges of the limit shape, we prove that the joint distribution of suitably scaled 1st, 2nd, and so on rows of a Plancherel typical diagram converges to the corresponding distribution for eigenvalues of random Hermitian matrices (given by the Airy kernel). This proves a conjecture due to Baik, Deift, and Johansson by methods different from the Riemann-Hilbert techniques used in their original papers math.CO/9810105 and math.CO/9901118 and from the combinatorial approach proposed by Okounkov in math.CO/9903176. Our approach is based on an exact determinantal formula for the correlation functions of the poissonized Plancherel measures involving a new kernel on the 1-dimensional lattice. This kernel is expressed in terms of Bessel functions and we obtain it as a degeneration of the hypergeometric kernel from the paper math.RT/9904010 by Borodin and Olshanski. Our asymptotic analysis relies on the classical asymptotic formulas for the Bessel functions and depoissonization techniques.
