Coherence invariant maps on order-3 symmetric tensors

Abstract

In 1940s, Hua established the fundamental theorems of geometry of rectangular matrices, symmetric matrices, skew-symmetric matrices, and hermitian matrices. In 1950s, Jacob generalized Hua's theorems to that of order-2 tensors and symmetric tensors. We extend Jacob's work to maps of order-3 symmetric tensors over $\mathbb{C}$ by proving that every surjective coherence invariant map on order-3 symmetric tensors over $\mathbb{C}$ is induced by a semilinear isomorphism apart from an additive constant.

Keywords:

• Symmetric tensors
• symmetric rank
• coherence invariant
• adjacency preserving

Acknowledgements

The author would like to thank the editor and the reviewer for their time and helpful comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 It can be shown that $\{u_1^{\otimes r},u_2^{\otimes r},u_3^{\otimes r}\}$ is linearly independent. Otherwise, with no loss of generality, say $u_3^{\otimes r}=\alpha u_1^{\otimes r}+\text{β}{u}_2^{\otimes r}$ for some $\alpha ,\text{β}\in \mathbb{C}$. Since $\{u_1^{\otimes r},u_2^{\otimes r}\}$ is linearly independent, so is $\{u_1,u_2\}$. This, together with the fact that $\mathrm{srank}(\alpha u_1^{\otimes r}+\text{β}{u}_2^{\otimes r})=\mathrm{srank}\left(u_3^{\otimes r}\right)=1$, implies that either $\alpha =0$ or $\text{β}=0$ by Lemma 2.1. This contradicts the linear independence of either $\{u_1^{\otimes r},u_3^{\otimes r}\}$ or of $\{u_2^{\otimes r},u_3^{\otimes r}\}$.

Additional information

Funding

The author would like to thank the Ministry of Higher Education (MoHE) Malaysia for the Fundamental Research Grant Scheme (FRGS/1/2019/STG06/UM/02/1) awarded to Kwa Kiam Heong.