Rank functions on rooted tree quivers

TL;DR

This paper studies the structure of the ring formed by indecomposable representations of rooted tree quivers, showing it is finitely generated and providing an explicit description, with implications for understanding quiver representations.

Contribution

It explicitly describes the reduced ring of indecomposable representations for rooted tree quivers and establishes its finite generation as a Z-module.

Findings

The ring R(Q)_{red} is finitely generated for rooted tree quivers.

An explicit description of R(Q)_{red} is provided.

Open problem proposed for future research.

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 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. We also present an open problem for future direction.