On polynomial integrals over the orthogonal group

TL;DR

This paper studies integrals over the orthogonal group involving matrix entries, revealing invariance properties and providing a general factorial-based formula using combinatorial methods.

Contribution

It introduces new invariance properties and a comprehensive factorial formula for polynomial integrals over the orthogonal group, advancing theoretical understanding.

Findings

Identified invariance properties of polynomial integrals over O(n)

Derived a general factorial formula for these integrals

Used combinatorial methods to establish key results

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.