Representations of quivers, their generalizations and invariants

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A quiver Q is a finite oriented graph. Its representation is a collection of vector spaces assigned to vertices together with linear mappings “along” arrows. To be able to consider bilinear forms on vertex vector spaces together with linear mappings between them the notion of mixed quiver settings was introduced. To define it we assume that vector spaces assigned to some vertices are dual to each other. All (mixed) representations with fixed dimensions of vector spaces constitute the space of representations H. The product of general linear groups G acts on vertex vector spaces as change of bases and thereby G acts on H. Orbits of this action correspond to isomorphic classes of representation of Q. Similarly, we can define the group G* for mixed representations. The first part of the research is dedicated to semisimple mixed representations and mixed representations with closed G*-orbits on H. The second part is dedicated to relations between semi-invariants, i.e., invariants with respect to the commutator of G, for the usual representations of quivers. We intend to apply this result to Littlewood-Richardson coefficients. The third part is dedicated to generators for quantum matrix invariants.