A geometric construction of Coxeter-Dynkin diagrams of bimodal singularities
نویسندگان
چکیده
We consider the Berglund-Hübsch transpose of a bimodal invertible polynomial and construct a triangulated category associated to the compactification of a suitable deformation of the singularity. This is done in such a way that the corresponding Grothendieck group with the (negative) Euler form can be described by a graph which corresponds to the Coxeter-Dynkin diagram with respect to a distinguished basis of vanishing cycles of the bimodal singularity.
منابع مشابه
Generalized Dynkin Diagrams and Root Systems and Their Folding Generalized Dynkin Diagrams and Root Systems and Their Folding
Graphs which generalize the simple or aane Dynkin diagrams are introduced. Each diagram deenes a bilinear form on a root system and thus a reeection group. We present some properties of these groups and of their natural \Coxeter element". The folding of these graphs and groups is also discussed, using the theory of C-algebras. Abstract Graphs which generalize the simple or aane Dynkin diagrams ...
متن کاملSemi-affine Coxeter-dynkin Graphs and G ⊆ Su 2 (c)
The semi-affine Coxeter-Dynkin graph is introduced, generalizing both the affine and the finite types. Semi-affine graphs. It is profitable to treat the so-called Coxeter-Dynkin diagrams as graphs. A classification of finite graphs with an adjacency matrix having 2 as the largest eigenvalue is made in a paper of John Smith [JHS]. It is in a combinatorial context and no reference is made to Coxe...
متن کاملGeneralized Dynkin diagrams and root systems and their folding
Graphs which generalize the simple or affine Dynkin diagrams are introduced. Each diagram defines a bilinear form on a root system and thus a reflection group. We present some properties of these groups and of their natural “Coxeter element”. The folding of these graphs and groups is also discussed, using the theory of C-algebras.
متن کاملCoxeter groups , quiver mutations and geometric manifolds Anna
We construct finite volume hyperbolic manifolds with large symmetry groups. The construction makes use of the presentations of finite Coxeter groups provided by Barot and Marsh and involves mutations of quivers and diagrams defined in the theory of cluster algebras. We generalize our construction by assigning to every quiver or diagram of finite or affine type a CW-complex with a proper action ...
متن کاملThe Dynkin Diagrams of Rational Double Points
Rational double points are the simplest surface singularities. In this essay we will be mainly concerned with the geometry of the exceptional set corresponding to the resolution of a rational double point. We will derive the classification of rational double points in terms of Dynkin diagrams.
متن کامل