Cycle-finite module categories

We describe the structure of module categories of finite dimensional algebras over an algebraically closed field for which the cycles of nonzero nonisomorphisms between indecomposable finite dimensional modules are finite (do not belong to the infinite Jacobson radical of the module category). Moreover, geometric and homological properties of these module categories are exhibited

On quiver Grassmannians and orbit closures for representation-finite algebras

We show that Auslander algebras have a unique tilting and cotilting module which is generated and cogenerated by a projective-injective; its endomorphism ring is called the projective quotient algebra. For any representation- nite algebra, we use the projective quotient algebra to construct desingularizations of quiver Grassmannians, orbit closures in representation varieties, and their desingularizations. This generalizes results of Cerulli Irelli, Feigin and Reineke

THE ZERO SET OF SEMI-INVARIANTS FOR EXTENDED DYNKIN QUIVERS

Abstract. We show that the set of common zeros ZQ,d of all semi-invariants vanishing at 0 on the variety rep(Q, d) of all representations with dimension vector d of an extended Dynkin quiver Q under the group GL(d) isacomplete intersection if d is “big enough”. In case rep(Q, d) doesnotcontainanopen GL(d)-orbit, which is the case not considered so far, the number of irreducible components of ZQ,d grows with d, exceptifQ is an oriented cycle. 1. Introduction an

B-Orbits of 2-Nilpotent Matrices and Generalizations

The orbits of the group Bn of upper-triangular matrices acting on 2-nilpotent complex matrices via conjugation are classified via oriented link patterns, generalizing A. Melnikov’s classification of the Bn-orbits on upper-triangular such matrices. The orbit closures as well as the “build-ing blocks ” of minimal degenerations of orbits are described. The classi-fication uses the theory of representations of finite-dimensional algebras. Furthermore, we initiate the study of the Bn-orbits on arbitrary nilpotent matrices.

Degeneration For Parabolic Group Actions In General Linear Groups

Let k be an algebraically closed field and V a finite dimensional k- space. Let GL(V ) be the general linear group of V and P a parabolic subgroup of GL(V ). Now P acts on its unipotent radical Pu and on pu = Lie Pu , the Lie algebra of Pu , via the adjoint action. More generally, we consider the action of P on the l-th member of the descending central series of pu denoted by p (l) u . All instances when P acts on p (l) u for l 0 with a finite number of orbits are known. In this note we give a complete description of the closure relations among the P -orbits on p (l) u in all these finite cases. There is a canonical bijection between the set of P -orbits on p (l) u and the set F (\Delta)(e) of isomorphism classes of \Delta--filtered modules of a particular dimension vector e of a certain quasi-hereditary algebra A(t; l). These isomorphism classes in turn are given by the orbits of the reductive group G(e) = Q GL(e i ) on the variety R(\Delta)(e) of all representations of..