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arXiv · 0802.3948

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

Benjamin Young·Jim Bryan
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Abstract

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. We find a relationship between the Donaldson-Thomas partition functions of the orbifold and its G-Hilbert scheme resolution. We formulate a crepant resolution conjecture for the Donaldson-Thomas theory of local orbifolds satisfying the Hard Lefschetz condition.

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Benjamin Young, Jim Bryan. 2008-07-02. Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds. https://doi.org/10.1215/00127094-2010-009

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