Invariants of quivers under the action of classical groups

Abstract

We consider a generalization of representations of quivers that can be derived from the ordinary representations of quivers by considering a product of arbitrary classical linear groups instead of a product of the general linear groups and by considering the dual action of groups on “vertex” vector spaces together with the usual action. A generating system for the corresponding algebra of invariants is found. In particular, semi-invariants of supermixed representations of quivers are established. As a consequence, a generating system for the algebra of SO ( n ) -invariants of several matrices is constructed over a field of characteristic different from 2. The proof uses the reduction to semi-invariants of mixed representations and the decomposition formula that generalizes Amitsur's formula for the determinant.