Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brouwer, P.W., Beenakker, C.W.J.: Diagrammatic method of integration over the unitary group, with applications to quantum transport in mesoscopic systems. J. Math. Phys. 37(10), 4904–4934 (1996)
Brézin, E., Hikami, S.: An extension of the Harish Chandra-Itzykson-Zuber integral. Commun. Math. Phys. 235(1), 125–137 (2003)
Biane, P.: Some properties of crossings and partitions. Discrete Math. 175(1–3), 41–53 (1997)
Biane, P.: Representations of symmetric groups and free probability. Adv. Math. 138(1), 126–181 (1998)
Bousquet-Mélou, M., Schaeffer, G.: Enumeration of planar constellations. Adv. in Appl. Math. 24(4):337–368 (2000)
Brauer, R.: On algebras which are connected with the semisimple continuous groups. Ann. Math. 38, 857–872 (1937)
Birman, J.S., Wenzl, H.: Braids, link polynomials and a new algebra. Trans. Amer. Math. Soc. 313(1), 249–273 (1989)
Collins, B.: Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability. Int. Math. Res. Not. 17:953–982 (2003)
Fulton, W.: Young tableaux. Volume 35 of London Mathematical Society Student Texts. Cambridge: Cambridge University Press, 1997
Gorin, T.: Integrals of monomials over the orthogonal group. J. Math. Phys. 43(6), 3342–3351 (2002)
Grood, C.: Brauer algebras and centralizer algebras for SO(2n,C). J. Algebra 222(2), 678–707 (1999)
Weingarten, D.: Asymptotic behavior of group integrals in the limit of infinite rank. J. Math. Phys. 19(5), 999–1001 (1978)
Wenzl, H.: On the structure of Brauer's centralizer algebras. Ann. of Math. (2), 128(1), 173–193 (1988)
Weyl, H.: The Classical Groups. Their Invariants and Representations. Princeton, NJ: Princeton University Press, 1939
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Y. Kawahigashi
B.C. is supported by a JSPS postdoctoral fellowship.
P.Ś. was supported by State Committee for Scientific Research (KBN) grant 2 P03A 007 23.
Rights and permissions
About this article
Cite this article
Collins, B., Śniady, P. Integration with Respect to the Haar Measure on Unitary, Orthogonal and Symplectic Group. Commun. Math. Phys. 264, 773–795 (2006). https://doi.org/10.1007/s00220-006-1554-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-006-1554-3