Complete invariants for simultaneous similarity
Always dealing with an arbitrary field we consider the variety $(k^{n\times n})^{p}$ under the action of $GL_{n}$ 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 $GL_{n}$-stable subsets and for each of these we construct finitely many invariant morphisms to $k$ separating the orbits. The complicated action of $GL_{n}$ by similarity is reduced to left multiplication of a product of $GL_{l_{i}}$'s on a product of $k^{l_{i}\times m_{i}}$'s. An analogous result holds for the left-right action of $GL_{m}\times GL_{n}$ on $(k^{m\times n })^{p}$ and more generally for all varieties of finite dimensional modules over some finitely generated algebra.