2008/07/31 by Ryan Kinser · 24 citations
Mathematics · #Abelian group #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics #Discrete mathematics #Functor #Homotopy and Cohomology in Algebraic Topology #Indecomposable module #Mathematics #Modulo #Nilpotent #Pure mathematics #Quiver #Ring (chemistry) #math.RA #math.RT #msc:15A69 #msc:16G20 #msc:19A22
paper · pdf · doi:10.1215/00127094-2010-006
published in Duke Mathematical Journal 152(1) (Duke University Press) · 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
arxiv created 2009/03/09 · openalex publication_date 2010/03/11 · arxiv updated 2019/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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