2011/11/02 by Mark James Parsons, Parsons, Mark James · 1 citation
Mathematics · #05E10 (Primary) 18E30 (Secondary) #13F60 #16G10 #16G20 #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:05E10 #msc:13F60 #msc:16G10 #msc:16G20 #msc:18E30
paper · pdf · doi:10.48550/arxiv.1111.0450
27 pages
arxiv created 2011/11/02 · openalex publication_date 2011/11/02 · arxiv updated 2011/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motivated by work of Barot, Geiss and Zelevinsky, we study a collection of Z-bases (which we call companion bases) of the integral root lattice of a root system of simply-laced Dynkin type. Each companion basis is associated with the quiver of a cluster-tilted algebra of the corresponding type. In type A, we establish that the dimension vectors of the finitely generated indecomposable modules over a cluster-tilted algebra may be obtained, up to sign, by expanding the positive roots in terms of any companion basis for the quiver of that algebra. This generalises part of Gabriel's Theorem. Also, we describe the relationship between different companion bases for the same quiver and show how to mutate a companion basis for a quiver to produce a companion basis for a mutation of that quiver.