Representations of quivers with applications to collections of matrices with fixed similarity types and sum zero

• Main Author: Kirk, Daniel
• Published: University of Leeds 2013
• Subjects: 512.482
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.616296

Given a collection of matrix similarity classes Cl, ... , Ck the additive matrix problem asks under what conditions do there exist matrices Ai E Cj for j = 1, ... , k such that Al + ' .. + Ak = O. This and similar problems have been examined under various guises in the literature. The results of Crawley-Boevey use the representation theory of quivers to link the additive matrix problem to the root systems of quivers. We relate the results of Crawley-Boevey to another partial solution offered by Silva et al. and develop some tools to interpret the solutions of Silva et al. in terms of root systems. The results of Crawley-Boevey require us to know the precise Jordan form of the similarity classes; we address the problem of invoking Crawley-Boevey's results when only the invariant polynomials are known and we are not permitted to use polynomial factorization. We use the machinery of symmetric quivers and symmetric representations to study the problem of finding symmetric matrix solutions to the additive matrix problem. We show the reflection functors, defined for representations of deformed preprojective algebras, can be defined for symmetric representations. We show every rigid irreducible solution to the additive matrix problem can be realized by symmetric matrices and we use algebraic geometry to show that in some circumstances there are solutions which cannot be realized by symmetric matrices. We show there exist symmetric representations of deformed preprojective algebras of foot dimension vectors when the underlying quiver is Dynkin or extended Dynkin of type An Of Dn.