October 13, 2025

A new formula of the determinant tensor with symmetries

Key Points

A new formula for the determinant tensor reveals insights into its symmetries and rank characteristics.
The derived formula asserts that crank(detn) ≤ rank(detn) ≤ n!2⌊(n−2)/2⌋ for n×n matrices in non-characteristic 2 fields.
A connection is established with prior work on the determinant tensor det4, expanding understanding in this area.
This research can further inform mathematical frameworks involving tensors and matrix theory.

Abstract

In this paper, we present a new formula of the determinant tensor detn for n×n matrices. In Ju et al. Finding tensor decompositions with sparse optimization. J. Korean Math. Soc. 2025; 62(1): 33–49, doi.org/10.4134/JKMS.j230579, the same authors found a new formula of 4×4 determinant tensor det4, which is available when the base field is not of characteristic 2. Considering some symmetries in that formula, we found a new formula so that Crank⁡(detn)≤rank⁡(detn)≤n!2⌊(n−2)/2⌋when the base field is not of characteristic 2.

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Cite This Study

Ju et al. (Sat,) studied this question.

synapsesocial.com/papers/68ec51e642911f61ef8b2572 https://doi.org/https://doi.org/10.1080/03081087.2025.2558606

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