Nontriviality of equations and explicit tensors in Cm⊗Cm⊗Cm of border rank at least 2m−2

For even (resp. odd) m, I show the Young-flattening equations for border rank of tensors in Cm⊗Cm⊗Cm of [7] are nontrivial up to border rank 2m−3 (resp. 2m−5) by writing down explicit tensors on which the equations do not vanish. Thus these tensors have border rank at least 2m−2 (resp. 2m−4). The result implies that there are nontrivial equations for border rank 2n2−n that vanish on the matrix multiplication tensor for n×n matrices. I also study the border rank of the tensors of [1] and the equations of [4]. I show the tensors T2k∈Ck⊗C2k⊗C2k of [1], despite having rank equal to 2k+1−1, have border rank equal to 2k. I show the equations for border rank of [4] are trivial in the case of border rank 2m−1 and determine their precise non-vanishing on the matrix multiplication tensor.