Stephen Lovett

Abstract:   This dissertation presents a generalization to quivers, namely symmetric quivers. With symmetric quivers defined in an appropriate way, one can extend the notion of a standard quiver representation and construct orthogonal and symplectic representations of the quiver. After presenting basic categorical and linear algebraic properties of orthogonal and symplectic representations of symmetric quivers, we study these representations' orbit closures. For any symmetric quiver we present a codimension formula for the orbit closure of representations while for symmetric quivers of finite type we construct desingularizations for the orbit closures. Focussing on the A₃ case, in which the orbit closures corresponds to orthogonal and symplectic analogues of determinantal ideals, we construct various resolutions of coordinate rings and analyze properties such as Cohen-Macaulayness, normality and rationality of singularities. As an application of our methods, we provide a simple characterization of the orthogonal or symplectic A₃ orbit closures that possess Gorenstein coordinate rings.