A geometric version of BGP reflection functors

Abstract

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 one of Reineke. 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.