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The problem of harmonic analysis on the infinitedimensional unitary group
 J. Funct. Anal. 205, no
"... Abstract. The infinite–dimensional unitary group U(∞) is the inductive limit of growing compact unitary groups U(N). In this paper we solve a problem of harmonic analysis on U(∞) stated in [Ol3]. The problem consists in computing spectral decomposition for a remarkable 4–parameter family of characte ..."
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Cited by 32 (10 self)
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Abstract. The infinite–dimensional unitary group U(∞) is the inductive limit of growing compact unitary groups U(N). In this paper we solve a problem of harmonic analysis on U(∞) stated in [Ol3]. The problem consists in computing spectral decomposition for a remarkable 4–parameter family of characters of U(∞). These characters generate representations which should be viewed as analogs of nonexisting regular representation of U(∞). The spectral decomposition of a character of U(∞) is described by the spectral measure which lives on an infinite–dimensional space Ω of indecomposable characters. The key idea which allows us to solve the problem is to embed Ω into the space of point configurations on the real line without 2 points. This turns the spectral measure into a stochastic point process on the real line. The main result of the paper is a complete description of the processes corresponding to our concrete family of characters. We prove that each of the processes is a determinantal point process. That is, its correlation functions have determinantal form with a certain kernel. Our kernels have a special ‘integrable ’ form and are expressed through the Gauss
Fredholm determinants, JimboMiwaUeno taufunctions, and representation theory
, 2001
"... The authors show that a wide class of Fredholm determinants arising in the representation theory of “big ” groups such as the infinite–dimensional unitary group, solve Painlevé equations. Their methods are based on the theory of integrable operators and the theory of Riemann–Hilbert problems. ..."
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Cited by 19 (5 self)
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The authors show that a wide class of Fredholm determinants arising in the representation theory of “big ” groups such as the infinite–dimensional unitary group, solve Painlevé equations. Their methods are based on the theory of integrable operators and the theory of Riemann–Hilbert problems.
On The Representations Of The Infinite Symmetric Group
, 1997
"... We classify all irreducible admissible representations of three Olshanski pairs connected to the infinite symmetric group S(1). In particular, our methods yield two simple proofs of the classical Thoma's description of the characters of S(1). ..."
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Cited by 8 (1 self)
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We classify all irreducible admissible representations of three Olshanski pairs connected to the infinite symmetric group S(1). In particular, our methods yield two simple proofs of the classical Thoma's description of the characters of S(1).
Combinatorial formula for Macdonald polynomials, Bethe Ansatz, and generic
, 2000
"... We give a direct proof of the combinatorial formula for interpolation Macdonald polynomials by introducing certain polynomials, which we call generic Macdonald polynomials, which depend on d additional parameters and specialize to all Macdonald polynomials of degree d. The form of these generic poly ..."
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Cited by 3 (1 self)
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We give a direct proof of the combinatorial formula for interpolation Macdonald polynomials by introducing certain polynomials, which we call generic Macdonald polynomials, which depend on d additional parameters and specialize to all Macdonald polynomials of degree d. The form of these generic polynomials is that of a Bethe eigenfunction and they imitate, on a more elementary level, the Rmatrix construction of quantum immanants.
REPRESENTATION THEORY AND RANDOM POINT PROCESSES
, 2004
"... Abstract. On a particular example we describe how to state and to solve the problem of harmonic analysis for groups with infinite–dimensional dual space. The representation theory for such groups differs in many respects from the conventional theory. We emphasize a remarkable connection with random ..."
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Cited by 1 (1 self)
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Abstract. On a particular example we describe how to state and to solve the problem of harmonic analysis for groups with infinite–dimensional dual space. The representation theory for such groups differs in many respects from the conventional theory. We emphasize a remarkable connection with random point processes that arise in random matrix theory. The paper is an extended version of the second author’s talk at the Congress.
ASYMPTOTICS OF MULTIVARIATE ORTHOGONAL POLYNOMIALS WITH HYPEROCTAHEDRAL SYMMETRY
, 2004
"... Abstract. We present a formula describing the asymptotics of a class of multivariate orthogonal polynomials with hyperoctahedral symmetry as the degree tends to infinity. The polynomials under consideration are characterized by a factorized weight function satisfying certain analyticity assumptions. ..."
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Abstract. We present a formula describing the asymptotics of a class of multivariate orthogonal polynomials with hyperoctahedral symmetry as the degree tends to infinity. The polynomials under consideration are characterized by a factorized weight function satisfying certain analyticity assumptions. As an application, the largedegree asymptotics of the KoornwinderMacdonald BCNtype multivariate AskeyWilson polynomials is determined. 1.
Factor Representations of Diffeomorphism Groups
, 2008
"... General semifinite factor representations of the diffeomorphism group of euclidean space are constructed by means of a canonical correspondence with the finite factor representations of the inductive limit unitary group. This construction includes the quasifree representations of the canonical comm ..."
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General semifinite factor representations of the diffeomorphism group of euclidean space are constructed by means of a canonical correspondence with the finite factor representations of the inductive limit unitary group. This construction includes the quasifree representations of the canonical commutation and anticommutation relations. To establish this correspondence requires a nonlinear form of complete positivity as developed by Arveson. We also compare the asymptotic character formula for the unitary group with the thermodynamic (N/V) limit construction for diffeomorphism group representations. 1
ENTROPY OF SCHUR–WEYL MEASURES
"... Abstract. Relative dimensions of isotypic components of N–th order tensor representations of the symmetric group on n letters give a Plancherel– type measure on the space of Young diagrams with n cells and at most N rows. It was conjectured by G. Olshanski that dimensions of isotypic components of t ..."
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Abstract. Relative dimensions of isotypic components of N–th order tensor representations of the symmetric group on n letters give a Plancherel– type measure on the space of Young diagrams with n cells and at most N rows. It was conjectured by G. Olshanski that dimensions of isotypic components of tensor representations of finite symmetric groups, after appropriate normalization, converge to a constant with respect to this family of Plancherel–type measures in the limit when N √ n converges to a constant.