Biquadratic spaces of length two

TL;DR

This paper classifies all universal biquadratic tensor spaces of length two, revealing seven families and contributing to the understanding of highly symmetrical tensor spaces and their applications.

Contribution

It provides the first classification of biquadratic spaces of length two, expanding the understanding of highly symmetrical tensor spaces.

Findings

Seven families of biquadratic spaces of length two identified

Established cases of the linear Ryll-Nardzewski theorem

Enhanced classification framework for symmetrical tensor spaces

Abstract

A tensor space is a vector space equipped with a finite collection of multilinear forms. The length of a tensor space is its length as a representation of its symmetry group. Infinite dimension tensor spaces of finite length are special, highly symmetrical objects. We classify the (universal) biquadratic spaces of length two; there are seven families of them. As a corollary, we establish some cases of the linear analog of the Ryll-Nardzewski theorem. We view this work as a first attempt to classify highly symmetrical tensor spaces.