#### 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.

#### Recommended Citation

McGregor-Dorsey, Zachary Strider, "Some properties of full heaps" (2013). *Mathematics Graduate Theses & Dissertations*. 28.

https://scholar.colorado.edu/math_gradetds/28