(λ, G)-critical matrices

AbstractThe problem of finding the conditions for equality of nonzero decomposable symmetrized tensors has been considered several times—i.e. given the vectors x1,…,xm, y1,…,ym of V, in what conditions one has x1∗⋯∗xm = y1∗⋯∗ym ≠ 0. We present here a necessary condition for (1), introducing the concept of (λ, G)-critical matrix. We present also some new results on the characterization of the set of the matrices A ∈ Mm×m(C) satisfying dGλ(AX) = dGλ(XA) for all m-by-m matrices X over C