2012/12/06 by Ingalls, Colin, Paquette, Charles, Thomas, Hugh · 1 citation
#05E10 #16G20 #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1212.1424
In this paper, we study the semi-stable subcategories of the category of representations of a Euclidean quiver, and the possible intersections of these subcategories. Contrary to the Dynkin case, we find out that the intersection of semi-stable subcategories may not be semi-stable. However, only a finite number of exceptions occur, and we give a description of these subcategories. Moreover, one can attach a simplicial fan in ℚn to any acyclic quiver Q, and this simplicial fan allows one to completely determine the canonical presentation of any element in ℤn. This fan has a nice description in the Dynkin and Euclidean cases: it is described using an arrangement of convex codimension-one subsets of ℚn, each such subset being indexed by a real Schur root or a set of quasi-simple objects. This fan also characterizes when two different stability conditions give rise to the same semi-stable subcategory.