Combinatorial aspects of orthogonal group integrals

Abstract

We study the integrals of type $I(a)={\int }_{O_n}\prod u_{ij}^{a_{ij}}du$, depending on a matrix $a\in M_{p\times q}(\mathbb{N})$, whose exact computation is an open problem. Our results are as follows: (1) an extension of the "elementary expansion" formula from the case $a\in M_{2\times q}(2\mathbb{N})$ to the general case $a\in M_{p\times q}(\mathbb{N})$, (2) the construction of the "best algebraic normalization" of $I(a)$, in the case $a\in M_{2\times q}(\mathbb{N})$, (3) an explicit formula for $I(a)$, for diagonal matrices $a\in M_{3\times 3}(\mathbb{N})$, (4) a modelling result in the case $a\in M_{1\times 2}(\mathbb{N})$, in relation with the Euler-Rodrigues formula. Most proofs use various combinatorial techniques.

arXiv comment34 pages