Narrow Results

The algebra of ${\mathrm{\text{GL}}}_n$-invariants of m-tuples of $n\times n$ matrices with respect to the action by simultaneous conjugation is a classical topic in case of infinite base field. On the other hand, in case of a finite field generators of polynomial invariants are not known even for a pair of $2\times 2$ matrices. Working over an arbitrary field we classified all ${\mathrm{\text{GL}}}_2$-orbits on m-tuples of $2\times 2$ nilpotent matrices for all $m>0$. As a consequence, we obtained a minimal separating set for the algebra of ${\mathrm{\text{GL}}}_2$-invariant polynomial functions of m-tuples of $2\times 2$ nilpotent matrices. We also described the least possible number of elements of a separating set for an algebra of invariant polynomial functions over a finite field.