Representation Theory Of Quivers Research Articles

Cluster algebras as Hall algebras of quiver representations

Recent articles have shown the connection between representation theory of quivers and the theory of cluster algebras. In this article, we prove that some cluster algebras of type A-D-E can be recovered from the data of the corresponding quiver representation category. This also provides some explicit formulas for cluster variables.

Regular points in system spaces

Representation theory of quivers is applied to system theory. As a result, the regular points in system spaces are characterized. Moreover, all prestable points in system spaces are described and another generic property of linear systems is introduced.

On Weakly Non-Negative Unit forms and Tame Algebras

In this paper we consider integral quadratic forms q: IP —>Z of the form q(x) = 2i x] + ^i<jqijX,Xj and call them unit forms. Such a unit form q is said to be weakly non-negative if q(x)^Q whenever Jt2*0, that is, whenever x^O for i = 1,..., n. Our main purpose is to formulate a criterion for weak non-negativity of unit forms. This question arises in various areas of representation theory in connection with the characterization of tame representation type: in the representation theory of quivers [17, 6], partially ordered sets [18, 19], certain classes of bimodules [8, 23, 1.3] and certain classes of algebras [20, 14, 24, 16]. For each of these structures Q, there is a unit form qn, naturally attached to Q, such that the weak non-negativity of qn is an equivalent (or at least a necessary) condition for Q to be of tame representation type. A class of algebras for which tame representation type is equivalent to the weak non-negativity of the associated unit form is described in the last section of this paper. A general criterion for weak non-negativity using determinants was given by Zeldich [28] (see also [21]), another criterion was obtained by de la Pena [21]. Our approach is similar to that in [11], where we gave a characterization of weakly positive unit forms based on the classification of the critical forms. Let us recall some notions. Given a unit form q: IP-*TL and a non-empty subset / = {ix < ... < im) of {1,..., n), we denote by qt the quadratic form qd, where d, Z —*• IT maps the /cth natural basis vector ek onto eik. Such a unit form q, is called the restriction of q to /. We call a unit form q on IP a hypercritical form if q, is weakly non-negative for every proper subset /<={1,..., n), though q itself is not. In these terms, a unit form q is weakly non-negative if and only if q admits no hypercritical restriction. So we have to classify the hypercritical forms. In § 1 we show that they can be obtained by certain elementary transformations from the unit forms naturally attached to the hyperbolic graphs of Fig. 1 on p. 50. In particular, there are only finitely many hypercritical forms. However, their number is large. By a simple process we reduce the problem to the classification of the stretched hypercritical forms. In § 2 we show that, except for a few of them, these are of the shape q(x) = q'(x') + xn -xn-iXn where q' = q{\,...,n-D is critical. (In a similar way, the unit forms attached to the hyperbolic graphs Am, Dm, Em correspond to those

Elementary construction of the quiver of the Mackey algebra for groups with cyclic normal p-sylow subgroup

Through recent investigations of J. Thrvenaz and P.J. Webb [4-8], the concept of Mackey functors, being originally introduced by J. A. Green [3] and studied by A.W.M. Dress, T. Yoshida, H. Sasaki, and others, has regained considerable interest. An important reason for this is ~i new interpretation of a conjecture of Aiperin in terms of Mackey functors in [8] using results of [7]. Given a commutative ring ,4, by its definition, it is immediate that the category of Mackey functors over .4 for a finite group G is an abelian category, even that it is the category of ,4-representations of a quiver with relations [8]. Namely a Mackey functor for G associates to each subgroup of G a ,(-module and moreover certain /'-linear maps between these spaces satisfying certain relations involving G. Thus a Mackey functor for G can be interpreted as a module over a path algebra of a quiver with relations, the so-called Mackey algebra [6, 9]. The main difference to the usual representation theory of quivers, and thus the main problem of interpreting Mackey functors as such representations, lies in the fact that the maps between these spaces do not lie in the radical of this path algebra, and this path algebra is not basic in general. So it is a natural problem to compute the quiver in the usual sense for the category of Mackey functors for a group G in particular in the case where ,4 is a field of characteristic p dividing the order of G. The question of when there are up to isomorphism only finitely many indecomposable Mackey functors for G led to recent investigations of Thrvenaz and Webb solving this problem in general, and it also led to the investigations we present in this paper which presents in a special situation, namely if G has a normal cyclic p-Sylow subgroup, an explicit computation of the quiver of the now so-called Mackcy algebra. By definition, the Mackey algebra is the basic ,(-algebra whose category of finitely generated

On normal subgroups of direct products

We investigate the equivalence classes of normal subdirect products of a product of free groups Fn1 × … × Fnk under the simultaneous equivalence relations of commensurability and conjugacy under the full automorphism group. By abelianisation, the problem is reduced to one in the representation theory of quivers of free abelian groups. We show there are infinitely many such classes when k≧3, and list the finite number of classes when k = 2.

A wild quiver in linear systems theory

We study the action α: (( T, L), ( A, B)) ↦ ( T -1 AT, T -1 BL) of the group Gl n ( K) × Gl m ( K) of linear state and input transformations on the space U n,m of control systems ( A, B) ϵ K n×n × K n×m with n different eigenvalues, K an algebraically closed field. This action is of interest in systems theory as well as the representation theory of quivers. We construct a canonical form for α on U n,m and determine completely its stabilizer subgroups. The results are based on an analysis of the scaling action (( D,E), B) ↦ D -1 BE of the group of diagonal matrices D n ( K) × D m ( K) on K n×m . We construct a canonical form for this action by graph-theoretic methods and determine the adherence order of the underlying stratification of K n× m .

On the stabilizer subgroup of a fair of matrices

Let ( F,G) be a pair of matrices defined over an arbitrary field, Fn × n, Gn × m. Consider the natural action of GL n x GL m on this pair given by ( F,G) ↦ ( gFg -1, gGh -1), where ( g,h) ∈ GL n × GL m . This action is of interest in system theory as well as the representation theory of quivers. In this paper we study the stabilizer subgroup of this action stab( F,G), i.e. {(g,h) ∈ GL n x GL m|gFg -1 = F,gGh -1 = G} .

Knots-quivers correspondence

We introduce and explore the relation between knot invariants and quiver representation theory, which follows from the identification of quiver quantum mechanics in D-brane systems representing knots. We identify various structural properties of quivers associated to knots, and identify such quivers explicitly in many examples, including some infinite families of knots, all knots up to 6 crossings, and some knots with thick homology. Moreover, based on these properties, we derive previously unknown expressions for colored HOMFLY-PT polynomials and superpolynomials for various knots. For all knots, for which we identify the corresponding quivers, the LMOV conjecture for all symmetric representations (i.e. integrality of relevant BPS numbers) is automatically proved.