On polynomial integrals over the orthogonal group

Abstract

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.

20 pages

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