2007/03/12 by Alexander Kirillov, Alexander Kirillov Jr., Kirillov, Alexander +2
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT) #math.CO #math.RT
paper · pdf · doi:10.48550/arxiv.math/0703361
27 pages, 10 figures. v2: Added new sections relating our results to the theory of quiver representations
openalex publication_date 2007/03/12 · arxiv created 2007/05/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we show that for a simply-laced root system a choice of C gives rise to a natural construction of the Dynkin diagram, in which vertices of the diagram correspond to C-orbits in R; moreover, it gives an identification of R with a certain subset Ihat of I x Z2h, where h is the Coxeter number. The set Ihat has a natural quiver structure; we call it the periodic Auslander-Reiten quiver. This gives a combinatorial construction of the root system associated with the Dynkin diagram I: roots are vertices of Ihat, and the root lattice and the inner product admit an explicit description in terms of Ihat. Finally, we relate this construction to the theory of quiver representations.