A Geometric Version of Bgp Reflection Functors

Quiver Grassmannians and quiver flags are natural generalisations of usual Grassmannians and flags. They arise in the study of quiver representations and Hall algebras. In general, they are projective varieties which are neither smooth nor irreducible. We use a scheme theoretic approach to calculate their tangent space and to obtain a dimension estimate similar to the one of Reineke in [Rei02]. Using this we can show that if there is a generic representation, then these varieties are smooth and irreducible. If we additionally have a counting polynomial we deduce that their Euler characteristic is positive and that the counting polynomial evaluated at zero yields one. After having done so, we introduce a geometric version of BGP reflection functors which allows us to deduce an even stronger result about the constant coefficient of the counting polynomial. We use this to obtain an isomorphism between the Hall algebra at q = 0 and Reineke’s generic extension monoid in the Dynkin case. One of the first non-trivial varieties one studies when learning algebraic geometry is the Grassmannian. It consists of r-dimensional subspaces of a fixed d-dimensional vector space V . The (vector space) Grassmannian is smooth and irreducible, has a nice functor of points and the tangent space at an r-dimensional subspace U of V is given by linear maps from U to V/U . A first generalisation of this variety is the flag variety, which consists of a filtration of V by subvector spaces of fixed dimension vectors. Although the vector space Grassmannian already leads to interesting geometrical problems, it is still very easy. A generalisation is given by taking quiver Grassmannians. A quiver Q is an oriented graph and a k-representation of Q is given by assigning a k-vector space to each vertex of Q and a k-linear map between these vector spaces to each arrow of Q. The dimension vector of a finite dimensional krepresentation M is the tuple of dimensions of the vector spaces at the vertices. For a k-representation M the quiver Grassmannian consists of subrepresentations of M of dimension vector d. In general, this scheme is much more complex, or interesting, as the vector space Grassmannian. It is generally neither smooth nor reduced nor irreducible. Studying the quiver Grassmannian was first done by Schofield [Sch92] when he introduced a calculus of Schur roots. Recently it received attention for example by Caldero and Chapoton [CC06] when they showed that its Euler characteristic gives coefficients in the cluster algebra. Caldero and Reineke [CR] showed that the Euler characteristic is positive provided that the quiver is acyclic and that the representation is rigid by using Lusztig’s pervese sheaf approach [Lus93] to quantum groups. There is also the natural generalisation to quiver flags, but up to now there has been no work investigating geometric properties of those. 2000 Mathematics Subject Classification. 16G20, 14L30. 1 ar X iv :0 90 8. 42 44 v1 [ m at h. R T ] 2 8 A ug 2 00 9