Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds

Understand

We derive two multivariate generating functions for three-dimensional Young diagrams (also called plane partitions).

• The variables correspond to a colouring of the boxes according to a finite Abelian subgroup G of SO(3).
• We use the vertex operator methods of Okounkov--Reshetikhin--Vafa for the easy case G = Z/n; to handle the considerably more difficult case G=Z/2 x Z/2, we will also use a refinement of the author's recent q--enumeration of pyramid partitions.
• In the appendix, we relate the diagram generating functions to the Donaldson-Thomas partition functions of the orbifold C^3/G.