Let Q be a quiver. M. Reineke and A. Hubery investigated the connection
between the composition monoid, as introduced by M. Reineke, and the generic
composition algebra, 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 the composition algebra at q=0 is isomorphic to the monoid
algebra of the composition monoid. Moreover, if Q is an acyclic, extended
Dynkin quiver, we show that there exists an epimorphism from the composition
algebra at q=0 to the monoid algebra of the composition monoid, and we describe
its non-trivial kernel.
Our main tool is a geometric version of BGP reflection functors on quiver
Grassmannians and quiver flags, that is varieties consisting of filtrations of
a fixed representation by subrepresentations of fixed dimension vectors. These
functors enable us to calculate various structure constants of the composition
algebra.
Moreover, we investigate geometric properties of quiver flags and quiver
Grassmannians, and show that under certain conditions, quiver flags are
irreducible and smooth. If, in addition, we have a counting polynomial, these
properties imply the positivity of the Euler characteristic of the quiver flag.Comment: 111 pages, doctoral thesis University of Paderborn (2009
