Communications in Mathematical Physics

, Volume 264, Issue 3, pp 773–795 | Cite as

Integration with Respect to the Haar Measure on Unitary, Orthogonal and Symplectic Group

  • Benoît CollinsEmail author
  • Piotr Śniady


We revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the Haar measure on the unitary group U(d). The previous result provided exact formulas only for 2d bigger than the degree of the integrated polynomial and we show that these formulas remain valid for all values of d. Also, we consider the integrals of polynomial functions on the orthogonal group O(d) and the symplectic group Sp(d). We obtain an exact character expansion and the asymptotic behavior for large d. Thus we can show the asymptotic freeness of Haar-distributed orthogonal and symplectic random matrices, as well as the convergence of integrals of the Itzykson–Zuber type.


Neural Network Statistical Physic Complex System Asymptotic Behavior Nonlinear Dynamics 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Department of Mathematics, Graduate School of ScienceKyoto UniversityKyotoJapan
  2. 2.Institute of MathematicsUniversity of WroclawWroclawPoland

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