Resolutions of defining ideals of orbit closures for quivers of type $A_3$

Abstract

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 so-called $1$-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.