Reconstruction Algebras of Type D (I)

TL;DR

This paper describes the structure of reconstruction algebras for dihedral groups, simplifying their relations and connecting their complexity to the toric case, thus aiding in understanding minimal resolutions.

Contribution

It provides an explicit description of reconstruction algebras of type D for dihedral groups, showing most relations derive from type A algebras, simplifying their analysis.

Findings

Most relations in the reconstruction algebra are derived from type A relations.

The problem of understanding minimal resolutions reduces to the toric case.

Explicit descriptions are provided for dihedral groups with rank one special CM modules.

Abstract

This is the second in a series of papers which give an explicit description of the reconstruction algebra as a quiver with relations; these algebras arise naturally as geometric generalizations of preprojective algebras of extended Dynkin diagrams. This paper deals with dihedral groups G=D_{n,q} for which all special CM modules have rank one, and we show that all but four of the relations on such a reconstruction algebra are given simply as the relations arising from a reconstruction algebra of type A. As a corollary, the reconstruction algebra reduces the problem of explicitly understanding the minimal resolution (=G-Hilb) to the same level of difficulty as the toric case.