Combinatorial aspects of orthogonal group integrals

TL;DR

This paper advances the understanding of integrals over orthogonal groups by extending formulas, constructing normalizations, and providing explicit solutions for specific matrix cases, using combinatorial methods.

Contribution

It generalizes the elementary expansion formula for all integer matrices, introduces a normalized form, and derives explicit formulas for particular matrix types.

Findings

Extended elementary expansion to general matrices

Constructed algebraic normalization for specific cases

Derived explicit formulas for diagonal matrices

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.