Complete invariants for simultaneous similarity

Abstract

Always dealing with an arbitrary field we consider the variety $(k^{n\times n}{)}^p$ under the action of $G{L}_n$ by simultaneous similarity. We define discrete and continuous invariants which completely determine the orbits. The discrete invariants induce a disjoint decomposition of the variety into finitely many locally closed $G{L}_n$-stable subsets and for each of these we construct finitely many invariant morphisms to $k$ separating the orbits. The complicated action of $G{L}_n$ by similarity is reduced to left multiplication of a product of $G{L}_{l_i}$'s on a product of $k^{l_i\times m_i}$'s. An analogous result holds for the left-right action of $G{L}_m\times G{L}_n$ on $(k^{m\times n}{)}^p$ and more generally for all varieties of finite dimensional modules over some finitely generated algebra.

12 pages, to appear in "Advances in Mathematics"