The Annihilating Ideal of the Fisher Integral

TL;DR

This paper characterizes the differential operators that annihilate the Fisher integral on the special orthogonal group, providing explicit generators and proving maximality of the ideal, with applications to integrals over matrix groups.

Contribution

It explicitly constructs the annihilating ideal of the Fisher integral on SO(n) and proves its maximality, offering new methods for differential operator analysis on matrix integrals.

Findings

Explicit set of differential operators generating the annihilating ideal.

Proof that the ideal is a maximal left ideal.

New approach for differential operators annihilating Fisher integrals.

Abstract

In this paper, we discuss a system of differential equations for the Fisher integral on the special orthogonal group. Especially, we explicitly give a set of linear differential operators which generates the annihilating ideal of the Fisher integral, and we prove that the annihilating ideal is a maximal left ideal of the ring of differential operators with polynomial coefficients. Our proof is given by a discussion concerned with an annihilating ideal of a Schwartz distribution associated with the Haar measure on the special orthogonal group. We also give differential operators annihilating the Fisher integral for the diagonal matrix by a new approach.