A notion of a mixed representation of a quiver can be derived from ordinary quiver representation by considering the dual action of groups on "vertex" vector spaces together with their usual action. A generating system for the algebra of semi-invariants of mixed representations of a quiver is determined. This is done by reducing the problem to the case of bipartite quivers of a special form and using a function DP on three matrices, which is a mixture of the determinant and two pfaffians.
Let R n,d be the ring of invariants of d-tuples of n × n matrices under the simultaneous conjugation action of the general linear group. A minimal generating system and a homogeneous system of parameters of R 3,3 are determined. Homogeneous systems of parameters of R 3,2 , R 4,2 are also pointed out.
A basis for a relatively free associative algebra with the identity x 3 = 0 over a field of an arbitrary characteristic is found. As an application, a minimal generating system for the 3 × 3 matrix invariant algebra is determined.
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