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A few remarks on colour–flavour transformations, truncations of random unitary matrices, Berezin reproducing kernels and Selberg-type integrals

Yan V Fyodorov1,2 and Boris A Khoruzhenko3

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We investigate diverse relations of the colour–flavour transformations (CFT) introduced by Zirnbauer in 'Supersymmetry for systems with unitary disorder: circular ensembles' (1996 J. Phys. A: Math. Gen. 29 7113–36) to various topics in random matrix theory and multivariate analysis, such as measures on truncations of unitary random matrices, Jacobi ensembles of random matrices, Berezin reproducing kernels and a generalization of the Selberg integral due to Kaneko, Kadell and Yan involving the Schur functions. Apart from suggesting explicit formulae for bosonic CFT for the unitary group in the range of parameters beyond that in Zirnbauer's paper we also suggest an alternative variant of the transformation with integration going over an unbounded domain of a pair of Hermitian matrices. The latter makes possible the evaluation of certain averages in random matrix theory.


PACS

11.25.Hf Conformal field theory, algebraic structures

11.25.Yb M theory

02.10.Yn Matrix theory

02.30.Cj Measure and integration

MSC

15A57 Other types of matrices (Hermitian, skew-Hermitian, etc.)

28C10 Set functions and measures on topological groups, Haar measures, invariant measures (See also 22Axx, 43A05)

15A52 Random matrices

Subjects

Mathematical physics

Particle physics and field theory

Dates

Issue 4 (26 January 2007)

Received 17 October 2006, in final form 27 November 2006

Published 9 January 2007



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    Yan V Fyodorov and Boris A Khoruzhenko 2007 J. Phys. A: Math. Theor. 40 669

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