A new formula of the determinant tensor with symmetries

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.