Rank Functions on Rooted Tree Quivers

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 will 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.