For a finite abelian group G ⊂ GL(n, k), we describe the coherent component Yθ of the moduli space Mθ of θ-stable McKay quiver representations. This is a not-necessarily-normal toric variety that admits a projective birational morphism Yθ → Ak /G obtained by variation of GIT quotient. As a special case, this gives a new construction of Nakamura’s G-Hilbert scheme Hilb that avoids the (typically...

We give four positive formulae for the (equioriented type A) quiver polynomials of Buch and Fulton [BF99]. All four formulae are combinatorial, in the sense that they are expressed in terms of combinatorial objects of certain types: Zelevinsky permutations, lacing diagrams, Young tableaux, and pipe dreams (also known as rc-graphs). Three of our formulae are multiplicity-free and geometric, mean...

We define and study the category Rep(Q, F1) of representations of a quiver in Vect(F1) the category of vector spaces ”over F1”. Rep(Q, F1) is an F1–linear category possessing kernels, co-kernels, and direct sums. Moreover, Rep(Q, F1) satisfies analogues of the Jordan-Hölder and Krull-Schmidt theorems. We are thus able to define the Hall algebra HQ of Rep(Q, F1), which behaves in some ways like ...

We investigate the cluster-tilted algebras of finite representation type over an algebraically closed field. We give an explicit description of the relations for the quivers for finite representation type. As a consequence we show that a (basic) cluster-tilted algebra of finite type is uniquely determined by its quiver. Also some necessary conditions on the shapes of quivers of cluster-tilted a...

The aim of these notes is to report some new developments on the problem of describing all possible shapes of the connected components of the Auslander-Reiten quiver ΓA of an artin algebra A. The problem is interesting since the shapes of these components carry some important information of the module category of A. For instance the algebra A is hereditary if and only if ΓA has a connected comp...

In [1],the authors defined algebra homomorphisms from the dual RingelHall algebra of certain hereditary abelian categoryA to an appropriate q-polynomial algebra. In the case that A is the representation category of an acyclic quiver, we give an alternative proof by using the cluster multiplication formulas in [9]. Moreover, if the underlying graph of Q is bipartite and the matrix B associated t...

Let Q be a quiver. M. Reineke and A. Hubery investigated the connection between the composition monoid CM(Q), as introduced by M. Reineke, and the generic composition algebra Cq(Q), as introduced by C. M. Ringel, specialised at q = 0. In this thesis we continue their work. We show that if Q is a Dynkin quiver or an oriented cycle, then C0(Q) is isomorphic to the monoid algebra QCM(Q). Moreover,...

We unify aspects of the equivariant geometry type D quiver representation varieties, double Grassmannians, and symmetric varieties G L ( a + b ) / × ; in particular we translate results about singularities orbit closures, combinatorics closure containment, torus K -theory between these three families. These are all obtained from our generalization construction Zelevinsky for A quivers to settin...