Momentum map images of representation spaces of quivers

Properness of moment maps for representations of quivers

In this note, we give a necessary and sufficient condition for the properness of moment maps for representations of quivers. We show that the moment map for the representations of a quiver is proper if and only if the quiver is acyclic, that is, it does not contain any oriented cycle.

箭图子表示与矩映射的 Horn 条件

Horn conditions for quiver subrepresentations and the moment map

本文通过归纳条件刻画一般箭图表示子表示的 Schubert 位置，推广了 Belkale 的 Schubert 簇相交准则并细化 Schofield 的维数向量刻画，从而导出相应矩锥的 Horn 型不等式。

Moment map flows and the Hecke correspondence for quivers

In this paper we investigate the convergence properties of the upwards gradient flow of the norm-square of a moment map on the space of representations of a quiver. The first main result gives a necessary and sufficient algebraic criterion for a complex group orbit to intersect the unstable set of a given critical point. Therefore we can classify all of the isomorphism classes which contain an initial condition that flows up to a given critical point. As an application, we then show that Nakajima's Hecke correspondence for quivers has a Morse-theoretic interpretation as pairs of critical point…

Morse theory of the moment map for representations of quivers

The results of this paper concern the Morse theory of the norm-square of the moment map on the space of representations of a quiver. We show that the gradient flow of this function converges, and that the Morse stratification induced by the gradient flow co-incides with the Harder-Narasimhan stratification from algebraic geometry. Moreover, the limit of the gradient flow is isomorphic to the graded object of the Harder-Narasimhan-Jordan-Hölder filtration associated to the initial conditions for the flow. With a view towards applications to Nakajima quiver varieties we construct explicit local …

Discrete dynamical systems from mutation-periodic quivers: examples and reduction

Several new mutation-periodic quivers of period higher than 1 are introduced as well as the associated discrete dynamical systems. The reduction of these systems is developed using either a presymplectic or a Poisson approach. The presymplectic approach leads to a reduced system whose iteration map is symplectic with respect to a log symplectic form. In the Poisson approach we build a Poisson structure invariant under the iteration map, leading to a reduced system whose variables are the Casimirs of such structure.

通过 Morse 复形上的杯积实现箭图簇的卷积

Convolution for quiver varieties via cup product on a Morse complex

本文通过 Morse 复形上的杯积编码拓扑信息，为 Nakajima 箭图簇的卷积表示构造提供新方法，并发展了弱横截性理论以处理矩映射范数平方情形。