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34
Stein's method and Plancherel measure of the symmetric group
, 2003
"... We initiate a Stein’s method approach to the study of the Plancherel measure of the symmetric group. A new proof of Kerov’s central limit theorem for character ratios of random representations of the symmetric group on transpositions is obtained; the proof gives an error term. The construction of ..."
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Cited by 17 (7 self)
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We initiate a Stein’s method approach to the study of the Plancherel measure of the symmetric group. A new proof of Kerov’s central limit theorem for character ratios of random representations of the symmetric group on transpositions is obtained; the proof gives an error term. The construction of an exchangeable pair needed for applying Stein’s method arises from the theory of harmonic functions on Bratelli diagrams. We also find the spectrum of the Markov chain on partitions underlying the construction of the exchangeable pair. This yields an intriguing method for studying the asymptotic decomposition of tensor powers of some representations of the symmetric group.
Stochastic dynamics related to Plancherel measure
 AMS Transl.: Representation Theory, Dynamical Systems, and Asymptotic Combinatorics
"... Dedicated to A. M. Vershik on the occasion of his 70th birthday Abstract. 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 intersectio ..."
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Cited by 16 (7 self)
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Dedicated to A. M. Vershik on the occasion of his 70th birthday Abstract. 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 C in the quadrant, these Young diagrams form a Markov process ΛC with continuous time. We also describe ΛC in terms of jump rates. Our main result is the computation of the dynamical correlation functions of such Markov processes and their bulk and edge scaling limits.
An explicit form for Kerov’s character polynomials
 Trans. Amer. Math. Soc
, 2005
"... Abstract. Kerov considered the normalized characters of irreducible representations of the symmetric group, evaluated on a cycle, as a polynomial in free cumulants. Biane has proved that this polynomial has integer coefficients, and made various conjectures. Recently, ´ Sniady has proved Biane’s co ..."
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Cited by 15 (4 self)
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Abstract. Kerov considered the normalized characters of irreducible representations of the symmetric group, evaluated on a cycle, as a polynomial in free cumulants. Biane has proved that this polynomial has integer coefficients, and made various conjectures. Recently, ´ Sniady has proved Biane’s conjectured explicit form for the first family of nontrivial terms in this polynomial. In this paper, we give an explicit expression for all terms in Kerov’s character polynomials. Our method is through Lagrange inversion. 1.
Action of Coxeter groups on mharmonic polynomials
 and KZ equations, Preprint 2001, QA/0108012
"... Abstract. The Matsuo–Cherednik correspondence is an isomorphism from solutions of Knizhnik–Zamolodchikov equations to eigenfunctions of generalized Calogero–Moser systems associated to Coxeter groups G and a multiplicity function m on their root systems. We apply a version of this correspondence to ..."
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Cited by 12 (1 self)
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Abstract. The Matsuo–Cherednik correspondence is an isomorphism from solutions of Knizhnik–Zamolodchikov equations to eigenfunctions of generalized Calogero–Moser systems associated to Coxeter groups G and a multiplicity function m on their root systems. We apply a version of this correspondence to the most degenerate case of zero spectral parameters. The space of eigenfunctions is then the space Hm of mharmonic polynomials, recently introduced in [11]. We compute the Poincaré polynomials for the space Hm and for its isotypical components corresponding to each irreducible representation of the group G. We also give an explicit formula for mharmonic polynomials of lowest positive degree in the Sn case. 1.
method, Jack measure, and the Metropolis algorithm
 J. Combin. Theory Ser. A
"... Abstract: The one parameter family of Jackα measures on partitions is an important discrete analog of Dyson’s β ensembles of random matrix theory. Except for special values of α = 1/2,1,2 which have group theoretic interpretations, the Jackα measure has been difficult if not intractable to analyze. ..."
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Cited by 11 (6 self)
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Abstract: The one parameter family of Jackα measures on partitions is an important discrete analog of Dyson’s β ensembles of random matrix theory. Except for special values of α = 1/2,1,2 which have group theoretic interpretations, the Jackα measure has been difficult if not intractable to analyze. This paper proves a central limit theorem (with an error term) for Jackα measure which works for arbitrary values of α. For α = 1 we recover a known central limit theorem on the distribution of character ratios of random representations of the symmetric group on transpositions. The case α = 2 gives a new central limit theorem for random spherical functions of a Gelfand pair. The proof uses Stein’s method and has interesting ingredients: an intruiging construction of an exchangeable pair, properties of Jack polynomials, and work of Hanlon relating Jack polynomials to the
GAUSSIAN FLUCTUATIONS OF CHARACTERS OF SYMMETRIC GROUPS AND OF YOUNG DIAGRAMS
, 2005
"... We study asymptotics of reducible representations of the symmetric groups Sq for large q. We decompose such a representation as a sum of irreducible components (or, alternatively, Young diagrams) and we ask what is the character of a randomly chosen component (or, what is the shape of a randomly c ..."
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Cited by 11 (3 self)
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We study asymptotics of reducible representations of the symmetric groups Sq for large q. We decompose such a representation as a sum of irreducible components (or, alternatively, Young diagrams) and we ask what is the character of a randomly chosen component (or, what is the shape of a randomly chosen Young diagram). Our main result is that for a large class of representations the fluctuations of characters (and fluctuations of the shape of the Young diagrams) are Gaussian; in this way we generalize Kerov’s central limit theorem. The considered class consists of representations for which characters almost factorize and it includes, for example, leftregular representation (Plancherel measure), tensor representations and this class is closed under induction, restriction, outer product and tensor product of representations. Our main tool in the proof is the method of genus expansion, well known from the random matrix theory.
ASYMPTOTICS OF CHARACTERS OF SYMMETRIC GROUPS, GENUS EXPANSION AND FREE PROBABILITY
, 2005
"... The convolution of indicators of two conjugacy classes on the symmetric group Sq is usually a complicated linear combination of indicators of many conjugacy classes. Similarly, a product of the moments of the Jucys–Murphy element involves many conjugacy classes with complicated coefficients. In thi ..."
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Cited by 9 (5 self)
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The convolution of indicators of two conjugacy classes on the symmetric group Sq is usually a complicated linear combination of indicators of many conjugacy classes. Similarly, a product of the moments of the Jucys–Murphy element involves many conjugacy classes with complicated coefficients. In this article we consider a combinatorial setup which allows us to manipulate such products easily: to each conjugacy class we associate a twodimensional surface and the asymptotic properties of the conjugacy class depend only on the genus of the resulting surface. This construction closely resembles the genus expansion from the random matrix theory. As the main application we study irreducible representations of symmetric groups Sq for large q. We find the asymptotic behavior of characters when the corresponding Young diagram rescaled by a factor q −1/2 converge to a prescribed shape. The character formula (known as the Kerov polynomial) can be viewed as a power series, the terms of which correspond to twodimensional surfaces with prescribed genus and we compute explicitly the first two terms, thus we prove a conjecture of Biane.
Infinitedimensional diffusions as limits of random walks on partitions, Probab. Theory Related Fields 144
, 2009
"... Abstract. Starting with finite Markov chains on partitions of a natural number n we construct, via a scaling limit transition as n → ∞, a family of infinite–dimensional diffusion processes. The limit processes are ergodic; their stationary distributions, the so–called z–measures, appeared earlier in ..."
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Cited by 8 (2 self)
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Abstract. Starting with finite Markov chains on partitions of a natural number n we construct, via a scaling limit transition as n → ∞, a family of infinite–dimensional diffusion processes. The limit processes are ergodic; their stationary distributions, the so–called z–measures, appeared earlier in the problem of harmonic analysis for the infinite symmetric group. The generators of the processes are explicitly described.
Veselov Generalised discriminants, deformed CalogeroMoserSutherland operators and superJack polynomials
 Adv. Math
"... Abstract. It is shown that the deformed CalogeroMoserSutherland (CMS) operators can be described as the restrictions on certain affine subvarieties of the usual CMS operators for infinite number of particles. The ideals of these varieties are shown to be generated by the Jack symmetric functions r ..."
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Cited by 7 (4 self)
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Abstract. It is shown that the deformed CalogeroMoserSutherland (CMS) operators can be described as the restrictions on certain affine subvarieties of the usual CMS operators for infinite number of particles. The ideals of these varieties are shown to be generated by the Jack symmetric functions related to the Young diagrams with special geometry. Combinatorial formulas for the related superJack and shifted superJack polynomials are given, i<j 1.