Combinatorial Aspects of Orthogonal Group Integrals

We study the integrals of type I(a) = ∫ On ∏ u aij ij du, depending on a matrix a ∈Mp×q(N), whose exact computation is an open problem. Our results are as follows: (1) an extension of the “elementary expansion” formula from the case a ∈M2×q(2N) to the general case a ∈Mp×q(N), (2) the construction of the “best algebraic normalization” of I(a), in the case a ∈M2×q(N), (3) an explicit formula for I(a), for diagonal matrices a ∈ M3×3(N), (4) a modelling result in the case a ∈ M1×2(N), in relation with the Euler-Rodrigues formula. Most proofs use various combinatorial techniques. Introduction An interesting open question, with several potential applications, is the exact computation of the polynomial integrals over the orthogonal group On. These integrals are best introduced in a “rectangular form”, as functions of a matrix a ∈Mp×q(N), as follows: