On polynomial integrals over the orthogonal group

We consider integrals of type ∫ On u1 11 . . . u an 1nu b1 21 . . . u bn 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. Introduction The computation of polynomial integrals over the orthogonal group On is a key problem in mathematical physics. These integrals are indeed known to appear in a wealth of concrete situations, coming from random matrices, lattice models, combinatorics. These integrals are best introduced in a “rectangular way”, as follows: