This paper concerns preprojective representations of a finite connected valued quiver without oriented cycles. For each such representation, an explicit formula in terms of the geometry of the quiver gives a unique, up to a certain equivalence, shortest (+)-admissible sequence such that the corresponding composition of reflection functors annihilates the representation. The set of equivalence c...

Abstract Quiver representations arise naturally in many areas across mathematics. Here we describe an algorithm for calculating the vector space of sections, or compatible assignments vectors to vertices, any finite-dimensional representation a finite quiver. Consequently, are able define and compute principal components with respect quiver representations. These solutions constrained optimisat...

A unitary (Euclidean) representation of a quiver is given by assigning to each vertex a unitary (Euclidean) vector space and to each arrow a linear mapping of the corresponding vector spaces. We recall an algorithm for reducing the matrices of a unitary representation to canonical form, give a certain description of the representations of canonical form, and reduce the problem of classifying Eu...

The representations of dimension vector α of the quiver Q can be parametrised by a vector space R(Q,α) on which an algebraic group Gl(α) acts so that the set of orbits is bijective with the set of isomorphism classes of representations of the quiver. We describe the semi–invariant polynomial functions on this vector space in terms of the category of representations. More precisely, we associate...

We define a special sort of weighted oriented graphs, signed quivers. Each of these yields a symmetric quiver, i.e., a quiver endowed with an involutive anti-automorphism and the inherited signs. We develop a representation theory of symmetric quivers, in particular we describe the indecom-posable symmetric representations. Their dimensions constitute root systems corresponding to certain symme...

(1) Study the class of hyper-Kähler manifolds which are hyper-Kähler reductions of finite dimensional quaternion vector spaces by products of unitary groups. (Probably it is better to assume that the action is linear.) Besides quiver varieties, hyper-Kähler toric varieties in the sense of Bielawski and Dancer [2] are such examples. When the quotients are nonsingular ? How much of geometric prop...

200 NOTICES OF THE AMS VOLUME 52, NUMBER 2 Introduction A quiver is just a directed graph.1 Formally, a quiver is a pair Q = (Q0,Q1) where Q0 is a finite set of vertices and Q1 is a finite set of arrows between them. If a ∈ Q1 is an arrow, then ta and ha denote its tail and its head, respectively. Let us fix a quiver Q and a base field K. Attach a finite dimensional vector space to each vertex ...

We study finite quasi-quantum groups in their quiver setting developed recently by the first author. We obtain a classification of finite-dimensional pointed Majid algebras of finite corepresentation type, or equivalently a classification of elementary quasi-Hopf algebras of finite representation type, over the field of complex numbers. By the Tannaka-Krein duality principle, this provides a cl...

We introduce the notion of “maximal rank type” for representations of quivers, which requires certain collections of maps involved in the representation to be of maximal rank. We show that real root representations of quivers are of maximal rank type. By using the maximal rank type property and universal extension functors we construct all real root representations of a particular wild quiver w...