# zbMATH — the first resource for mathematics

On polynomial integrals over the orthogonal group. (English) Zbl 1231.05282
Summary: We consider integrals of type $$\int_{O_n} u_{11}^{a_1} \cdots u_{1n}^{a_n}u^{b_1}_{21} \cdots u^{b_n}_{2n}du$$, with respect to the Haar measure on the orthogonal group. We establish several remarkable invariance properties satisfied by such integrals, by using combinatorial methods. We present as well a general formula for such integrals, as a sum of products of factorials.

##### MSC:
 5e+10 Combinatorial aspects of representation theory
##### Keywords:
orthogonal group; Haar measure; hyperspherical law
Full Text:
##### References:
 [1] Banica, T., The orthogonal Weingarten formula in compact form, Lett. math. phys., 91, 105-118, (2010) · Zbl 1187.60002 [2] Banica, T.; Collins, B.; Schlenker, J.-M., On orthogonal matrices maximizing the 1-norm, Indiana univ. math. J., 59, 3, 839-856, (2010) · Zbl 1228.15013 [3] Banica, T.; Collins, B.; Zinn-Justin, P., Spectral analysis of the free orthogonal matrix, Int. math. res. not. IMRN, 17, 3286-3309, (2009) · Zbl 1179.46056 [4] Banica, T.; Goswami, D., Quantum isometries and noncommutative spheres, Comm. math. phys., 298, 343-356, (2010) · Zbl 1204.58004 [5] Collins, B.; Matsumoto, S., On some properties of orthogonal Weingarten functions, J. math. phys., 50, 1-18, (2009) [6] Collins, B.; Śniady, P., Integration with respect to the Haar measure on the unitary, orthogonal and symplectic group, Comm. math. phys., 264, 773-795, (2006) · Zbl 1108.60004 [7] Curran, S., Quantum rotatability, Trans. amer. math. soc., 362, 4831-4851, (2010) · Zbl 1203.46043 [8] Diaconis, P.; Shahshahani, M., On the eigenvalues of random matrices, J. appl. probab., 31, 49-62, (1994) · Zbl 0807.15015 [9] S. Matsumoto, J. Novak, Primitive factorizations, Jucys-Murphy elements, and matrix models, in: FPSAC 2010, in: Discrete Math. Theor. Comput. Sci., in press, arXiv:1005.0151. · Zbl 1374.05010 [10] Neretin, Y.A., Hua type integrals over unitary groups and over projective limits of unitary groups, Duke math. J., 114, 239-266, (2002) · Zbl 1019.43008 [11] Novak, J., Jucys-murphy elements and the unitary Weingarten function, Banach center publ., vol. 89, (2010), pp. 231-235 · Zbl 1219.05196 [12] Olshanskii, G.I., Unitary representations of infinite-dimensional pairs $$(G, K)$$ and the formalism of R. Howe, Adv. stud. contemp. math., 7, 269-463, (1990) [13] Petkovšek, M.; Wilf, H.S.; Zeilberger, D., $$a = b$$, (1996), A.K. Peters [14] Petz, D.; Réffy, J., Large deviation for the empirical eigenvalue density of truncated Haar unitary matrices, Probab. theory related fields, 133, 175-189, (2005) · Zbl 1076.60022 [15] Weingarten, D., Asymptotic behavior of group integrals in the limit of infinite rank, J. math. phys., 19, 999-1001, (1978) · Zbl 0388.28013 [16] Zinn-Justin, P., Jucys-murphy elements and Weingarten matrices, Lett. math. phys., 91, 119-127, (2010) · Zbl 1283.05269
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.