Applications of Representations of Quivers

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Abstract.This proposal is concerned with interactions between the representations of quivers, geometry and representation theory.The investigator will study the rings of semi-invariants of quivers and the combinatorial invariants they define. Special attention will be paid to the cone of weights of such rings and its relation to representation theory. In the special case of triple flag quivers this amounts to studying the cone defined by Klyachko inequalities. The principal investigator will also study the generalized quivers associated to reductive groups and their semi-invariants as well as the products of homogeneous spaces with finitely many orbits.Quivers are just oriented graphs. Their representations provide a convenient coding scheme allowing to study classification of objects arising in various areas of mathematics. The classification of matrices by their ranks and Jordan classificaton of endomorphisms of a vector space are the simplest examples. The more complicated ones include representations of classical groups and their generalizations to infinite dimensional algebras. Studying such classification problems, and conditions under which they can be explicitly solved is central in mathematical research. The investigator and his collaborators found that new interesting results can be obtained by using methods of invariant theory which were neglected before.