RESOLUTIONS OF DEFINING IDEALS OF ORBIT CLOSURES FOR QUIVERS OF TYPE A_3

摘要

We construct explicitly a minimal free resolution of the defining ideal of an orbit closure arising from a representation of the non-equioriented A_3 quiver. The resolution is a generalization of Lascoux’s resolution for determinantal ideals. The case of non-equioriented A_3 quiver is made special by the fact that, in this case, every orbit closure admits a socalled 1-step desingularization. Using the resolution we give a description of the minimal set of generators of the defining ideal. The resolution also allows us to read off some geometric properties of the orbit closure, like normality and Cohen- Macaulay. In addition, we give a characterization for the orbit closure to be Gorenstein.