Minimal singularities for representations of Dynkin quivers

We develop some reduction techniques for the study of singularities in orbit closures of finite dimensional modules. This enables us to classify all singularities occurring in minimal degenerations of representations of Dynkin quivers. They are all smoothly equivalent to the singularity at the zero-matrix inside thep×q-matrices of rank at most one.     

Locally semi-simple representations of quivers

We suggest a geometrical approach to the semi-invariants of quivers based on Luna's slice theorem and the Luna-Richardson theorem. The locally semi-simple representations are defined in this spirit but turn out to be connected with stable representations in the sense of GIT, Schofield's perpendicular categories, and Ringel's regular representations. As an application of this method we obtain an independent short proof of a theorem of Skowronski and Weyman about semi-invariants of the tame quivers.     

Normality of Marsden-Weinstein reductions for representations of quivers

 We prove that the Marsden-Weinstein reductions for the moment map associated to representations of a quiver are normal varieties. We give an application to conjugacy classes of matrices. Received: 10 August 2001 / Published online: 16 October 2002 Mathematics Subject Classification (2000): 16G20, 53D20, 14B05.     

Asymptotic degeneration of representations of quivers

We define asymptotic degeneration of nilpotent representations of an arbitrary finite quiver, using large tensor powers and small direct sums, and characterize this notion by a simple and effective criterion.     

Semi-invariants of mixed representations of quivers

A notion of a mixed representation of a quiver can be derived from ordinary quiver representation by considering the dual action of groups on "vertex" vector spaces together with their usual action. A generating system for the algebra of semi-invariants of mixed representations of a quiver is determined. This is done by reducing the problem to the case of bipartite quivers of a special form and using a function DP on three matrices, which is a mixture of the determinant and two pfaffians.     

On complexity of representations of quivers

It is shown that, given a representation of a quiver over a finite field, one can check in polynomial time whether it is absolutely indecomposable.     

Codimension two singularities for representations of extended Dynkin quivers

Let M and N be two representations of an extended Dynkin quiver such that the orbit of N is contained in the orbit closure and has codimension two. We show that the pointed variety is smoothly equivalent to a simple surface singularity of type A _n , or to the cone over a rational normal curve.     

On minimal disjoint degenerations for preprojective representations of quivers

We derive a root test for degenerations as described in the title. In the case of Dynkin quivers this leads to a conceptual proof of the fact that the codimension of a minimal disjoint degeneration is always one. For Euclidean quivers, it enables us to show a periodic behaviour. This reduces the classification of all these degenerations to a finite problem that we have solved with the aid of a computer. It turns out that the codimensions are bounded by two. Somewhat surprisingly, the regular and preinjective modules play an essential role in our proofs.

     

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 co-ordinates around the Morse strata and (under a technical hypothesis on the stability parameter) describe the negative normal space to the critical sets. Finally, we observe that the usual Kirwan surjectivity theorems in rational cohomology and integral K-theory carry over to this non-compact setting, and that these theorems generalize to certain equivariant contexts.     

On the Harder-Narasimhan filtration for finite dimensional representations of quivers

We prove that the Harder-Narasimhan filtration for an unstable finite dimensional representation of a finite quiver coincides with the filtration associated to the 1-parameter subgroup of Kempf, which gives maximal unstability in the sense of Geometric Invariant Theory for the corresponding point in the parameter space where these objects are parametrized in the construction of the moduli space.     

Topology of character varieties and representations of quivers

In Hausel et al. (2008) [10] we presented a conjecture generalizing the Cauchy formula for Macdonald polynomial. This conjecture encodes the mixed Hodge polynomials of the character varieties of representations of the fundamental group of a punctured Riemann surface of genus g. We proved several results which support this conjecture. Here we announce new results which are consequences of those in Hausel et al. (2008) [10].     

On spherical representations of quivers and generalized complexes

For any quiverQ we consider spherical representationsV ofQ such that the isomorphism class ofV is a spherical variety. We suggest an approach for classifying such representations for anyQ and obtain a classification forQ being an equioriented Dynkin diagramA _n. In particular, all complexes are spherical representations. We introduce a category of representations that we call generalized complexes and classify spherical generalized complexes. For the quivers that we call crumbly we prove that any spherical generalized complex has a polynomial algebra of covariants on the closure of its isomorphism class.Partially supported by INTAS-OPEN grant 97-1570 and RFFI grant 98-01-00598.     

t-structures in the derived category of representations of quivers

Given a finite quiver without oriented cycles, we describe a family of algebras whose module category has the same derived category as that of the quiver algebra. This is done in the more general setting oft-structures in triangulated categories. A completeness result is shown for Dynkin quivers, thus reproving a result of Happel [H].     

Almost split sequences in categories of representations of quivers

Let be a field and a connected quiver. In this note it is proved that the category of finite dimensional representations of over has almost split sequences if and only if either is without oriented cycles or consists of a single oriented cycle.

     

Relative invariants for representations of finite dimensional algebras

 Let M be a finite dimensional module over a finite dimensional basic K-algebra Λ, where K is an algebraically closed field. We associate with M a weight θ ^M (i.e. an element of the dual of the Grothendieck group of mod-Λ) in module theoretic terms. Let β be a dimension vector with θ ^M (β)=0. We generalize a construction of relative invariants of quivers due to Schofield [S] and define a relative invariant polynomial function d ^M _β on the variety of modules of dimension vector β, such that d ^M _β (N) = 0 for some module N if and only if there is a nonzero morphism from M to N. Assuming char (K) = 0, we conclude from the main result of Schofield-Van den Bergh [SV] that relative invariants of this form span all the spaces of relative invariants. To get algebra generators of the algebra of semi-invariants it is sufficient to take the d ^M _β with M indecomposable. Received: 31 July 2001