Counting coloured boxes

This thesis consists of the manuscripts of two research papers. In the first paper, we verify a recent conjecture of Kenyon/Szendroi by computing the generating function for pyramid partitions. Pyramid partitions are closely related to Aztec Diamonds; their generating function turns out to be the partition function for the Donaldson-Thomas theory of a non-commutative resolution of the conifold singularity {x₁ x₂ - x₃ x₄ = 0}⊂ C⁴. The proof does not require algebraic geometry; it uses a modified version of the domino (or dimer) shuffling algorithm of Elkies, Kuperberg, Larsen and Propp. In the second paper, 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). These generating functions turn out to be orbifold Donaldson-Thomas partition functions for the orbifold [C³/G]. We need only the vertex operator methods of Okounkov-Reshetikhin-Vafa for the easy case G = Zn; to handle the considerably more difficult case G = Z₂ x Z₂ , we will also use a refinement of the author's recent q-enumeration of pyramid partitions. In the appendix, written by Jim Bryan, we relate the diagram generating functions to the Donaldson-Thomas partition functions of the orbifold [C³/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.

Name thesis:degree_name

Doctor of Philosophy - PhD

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Mathematics

Grantor dc:publisher

University of British Columbia

Year dc:date

2008

Author dc:creator

• Young, Benjamin

Statement dc:rights

• Attribution-NonCommercial-NoDerivatives 4.0 International

Licence

• http://creativecommons.org/licenses/by-nc-nd/4.0/

Language dc:language

eng

Handle dc:identifier

http://hdl.handle.net/2429/731

OAI identifier oai:identifier

oai:circle.library.ubc.ca:2429/731