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 $\mathbb{Z}$-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.

42 pages, hyperlinked. Incorporates suggestions from an anonymous referee, notably a proof of Prop. 2 using sheaves, correction of a minor error in Prop. 32, and elimination of the assumption "K infinite" in several parts of the conclusion

References in corpus (1)

• The rank of a quiver representation

Cited by in corpus (10)

• Frobenius-Perron Theory of Representations of Quivers
• Quiver Bialgebras and Monoidal Categories
• Tree modules and counting polynomials
• Coefficient Quivers, ${\mathbb{F}}_1$-Representations, and Euler Characteristics of Quiver Grassmannians
• Idempotents in representation rings of quivers
• Tree normal forms for quiver representations
• K-polynomials of type A quiver orbit closures and lacing diagrams
• On quiver representations over ${\mathbb{F}}_1$
• Rank loci in representation spaces of quivers
• The Green rings of minimal Hopf quivers