Nontriviality of equations and explicit tensors in Cm⊗Cm⊗Cm of border rank at least 2m−2
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© 2014 Elsevier B.V. For even (resp. odd) m, I show the Young-flattening equations for border rank of tensors in Cm⊗Cm⊗Cm of  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  and the equations of . I show the tensors T2k∈Ck⊗C2k⊗C2k of , despite having rank equal to 2k+1-1, have border rank equal to 2k. I show the equations for border rank of  are trivial in the case of border rank 2m-1 and determine their precise non-vanishing on the matrix multiplication tensor.
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