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A few remarks on Colour-Flavour Transformations,truncations of random unitary matrices, Berezin reproducing kernels and Selberg type integrals

We investigate diverse relations of the colour-flavour transformations (CFT) introduced by Zirnbauer in \cite{Z1,Z2} 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 formulas for bosonic CFT for the unitary group in the range of parameters beyond that in \cite{Z2}, 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 evaluation of certain averages in random matrix theory.