Natural symmetric tensor norms

TL;DR

This paper introduces and classifies natural symmetric tensor norms for n-fold tensors, revealing six such norms for n≥3 and analyzing their associated polynomial ideals and algebra-preserving properties.

Contribution

It provides a complete classification of natural symmetric tensor norms for higher-order tensors and explores their algebraic and ideal-theoretic properties.

Findings

Exactly six natural symmetric tensor norms for n≥3

Four natural symmetric tensor norms for 2-fold tensors

Identification of polynomial ideals associated to these norms

Abstract

In the spirit of the work of Grothendieck, we introduce and study natural symmetric n-fold tensor norms. We prove that there are exactly six natural symmetric tensor norms for $n\ge 3$, a noteworthy difference with the 2-fold case in which there are four. We also describe the polynomial ideals associated to these natural symmetric tensor norms. Using a symmetric version of a result of Carne, we establish which natural symmetric tensor norms preserve Banach algebras.