Characterization of Gaussian Tensor Ensembles

TL;DR

This paper extends classical Gaussian ensemble characterizations to higher-order tensors, defining invariant polynomials and proving a Maxwell-type theorem for these tensor distributions.

Contribution

It introduces Gaussian tensor ensembles for symmetric, hermitian, and self-dual tensors, generalizing classical vector and matrix Gaussian ensembles.

Findings

Defined invariant polynomials for tensor ensembles

Proved a Maxwell-type theorem for Gaussian tensor distributions

Unified and extended classical ensemble characterizations

Abstract

The starting point of this work is a theorem due to Maxwell characterizing the distribution of a Gaussian vector with at least two coordinates. We define the Gaussian orthogonal, unitary and symplectic tensor ensembles for notions of real symmetric, hermitian and self-dual hermitian tensors which recover the classical vector and matrix Gaussian ensembles when the order is one and two. We give a complete family of invariant polynomials for orthogonal, unitary and symplectic transformations and we prove a Maxwell-type theorem for these Gaussian tensor distributions unifying and extending the ones known for vectors and matrices.