On polynomial integrals over the orthogonal group

Teodor Banica (1) , Benoit Collins (2, 3) , Jean-Marc Schlenker (4)

1 AGM - UMR 8088 - Analyse, Géométrie et Modélisation
2 Department of Mathematics and Statistics [Ottawa]
3 ICJ - Institut Camille Jordan
4 IMT - Institut de Mathématiques de Toulouse UMR5219

We consider integrals of type $\int_{O_n}u_{11}^{a_1}... u_{1n}^{a_n}u_{21}^{b_1}... u_{2n}^{b_n} du$, 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.

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]

Teodor Banica, Benoit Collins, Jean-Marc Schlenker. On polynomial integrals over the orthogonal group. J. Combin. Theory Ser. A, 2011, 118 (3), pp.778-795. ⟨10.1016/j.jcta.2010.11.015⟩. ⟨hal-00618945⟩