Date of Award

Spring 1-1-2013

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Richard M. Green

Second Advisor

Nathaniel Thiem

Third Advisor

Martin E. Walter

Fourth Advisor

J. M. Douglas

Fifth Advisor

Stephen R. Doty

Abstract

A full heap is a labeled infinite partially ordered set with labeling taken from the vertices of an underlying Dynkin diagram, satisfying certain conditions intended to capture the structure of that diagram. The notion of full heaps was introduced by R. Green as an affine extension of the minuscule heaps of J. Stembridge. Both authors applied these constructions to make observations of the Lie algebras associated to the underlying Dynkin diagrams. The main result of this thesis, Theorem 4.7.1, is a complete classification of all full heaps over Dynkin diagrams with a finite number of vertices, using only the general notion of Dynkin diagrams and entirely elementary methods that rely very little on the associated Lie theory. The second main result of the thesis, Theorem 5.1.7, is an extension of the Fundamental Theorem of Finite Distributive Lattices to locally finite posets, using a novel analogue of order ideal posets. We apply this construction in an analysis of full heaps to find our third main result, Theorem 5.5.1, an ADE classification of the full heaps over simply laced affine Dynkin diagrams.

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