Premium
Semi‐stable subcategories for Euclidean quivers
Author(s) -
Ingalls Colin,
Paquette Charles,
Thomas Hugh
Publication year - 2015
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdv002
Subject(s) - quiver , mathematics , subcategory , euclidean geometry , intersection (aeronautics) , pure mathematics , complete intersection , combinatorics , codimension , simplicial complex , geometry , engineering , aerospace engineering
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 Q n to any acyclic quiver Q , and this simplicial fan allows one to completely determine the canonical presentation of any element in Z n . This fan has a nice description in the Dynkin and Euclidean cases: it is described using an arrangement of convex codimension‐1 subsets of Q n , 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.