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 $R(Q{)}_{\mathrm{red}}$ is a finitely generated $\mathbb{Z}$-module (here $R(Q{)}_{\mathrm{red}}$ is the ring $R(Q)$ modulo the ideal of all nilpotent elements). We describe the ring $R(Q{)}_{\mathrm{red}}$ explicitly by studying functors from the category $\mathrm{rep}(Q)$ of representations of $Q$ over $K$ to the category of finite-dimensional $K$-vector spaces