Rank functions on rooted tree quivers

Abstract

The free abelian group R(Q) on the set of indecomposable representations of a quiver Q, over a field K, has a ring structure where the multiplication is given by the tensor product. We show that if Q is a rooted tree (an oriented tree with a unique sink), then the ring R(Q)red is a finitely generated Z-module (here R(Q)red is the ring R(Q) modulo the ideal of all nilpotent elements). We describe the ring R(Q)red explicitly by studying functors from the category rep(Q) of representations of Q over K to the category of finite-dimensional K-vector spaces