Complexity of matrix problems Original Research Article

In representation theory, the classification problem is called wild if it contains the problem of classifying pairs of matrices up to simultaneous similarity. We show in an explicit form that the last problem contains all classification matrix problems given by quivers or posets. Then we prove that this problem does not contain (but is contained in) the problem of classifying three-valent tensors. Hence, every wild classification problem given by a quiver or poset has the same complexity; moreover, a solution of one of them implies a solution of each of the remaining problems. The problem of classifying three-valent tensors is more complicated.