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Pizzetti Formulae for Stiefel Manifolds and Applications

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Abstract

Pizzetti’s formula explicitly shows the equivalence of the rotation invariant integration over a sphere and the action of rotation invariant differential operators. We generalize this idea to the integrals over real, complex, and quaternion Stiefel manifolds in a unifying way. In particular, we propose a new way to calculate group integrals and try to uncover some algebraic structures which manifest themselves for some well-known cases like the Harish-Chandra integral. We apply a particular case of our formula to an Itzykson–Zuber integral for the coset \({{{\rm SO}(4)/[{\rm SO}(2)\times {\rm SO}(2)]}}\). This integral naturally appears in the calculation of the two-point correlation function in the transition of the statistics of the Poisson ensemble and the Gaussian orthogonal ensemble in random matrix theory.

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Authors and Affiliations

  1. Department of Mathematical Analysis, Ghent University, Krijgslaan 281, 9000, Gent, Belgium

    Kevin Coulembier

  2. Mathematical Institute, Cologne University, Züpicher Str. 77, 50937, Cologne, Germany

    Kevin Coulembier

  3. Fakultät für Physik, Universität Bielefeld, 33501, Bielefeld, Germany

    Mario Kieburg

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  1. Kevin Coulembier
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  2. Mario Kieburg
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Coulembier, K., Kieburg, M. Pizzetti Formulae for Stiefel Manifolds and Applications. Lett Math Phys 105, 1333–1376 (2015). https://doi.org/10.1007/s11005-015-0774-x

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