Marti L. McClard, "Picturing Representations of Simple Lie Algebras of Rank Two," defended April 2000.

ABSTRACT:
The two known families of supporting graphs for the Gelfand-Tsetlin bases for the irreducible representations of the rank two simple Lie algebra A2 are presented as a guide for producing analogous distributive lattice supports for the irreducible representations of shape λ of the simple Lie algebra G2. Littelmann produced a set of tableaux of shape "6 x λ" that had the correct numbers of elements; i.e., the number of elements in his set of tableaux equaled the dimension of the corresponding representation of the Lie algebra G2. A translation of these objects into tableaux of shape λ is made and a method of constructing distributive lattices with several important combinatorial qualities from these translated G2 tableaux of shape λ is presented. Strong evidence is provided to support the claim that these Littelmann lattices are indeed supporting graphs for the irreducible representations of the Lie algebra G2.

L. Wyatt Alverson II, "Distributive Lattices and Representations of the Rank Two Simple Lie Algebras," defended July 2003.

ABSTRACT:
Three families of distributive lattices (so-called "semistandard lattices for A2, B2, and G2") which arise naturally in a certain algebraic context are considered. Two of these families of semistandard lattices (A2 and G2) were studied previously by McClard [M. McClard, Murray State University master's thesis, 2000]. From a combinatorial point of view, the lattices are quite striking and exhibit some pleasant symmetries and certain enumerative niceties. Evidence is presented suggesting that many of these lattices provide a suitable combinatorial environment for realizing linear representations of certain Lie algebras. This evidence includes results of McClard and others which led to a conjecture that all semistandard lattices could be used to realize Lie algebra representations. This conjecture has been confirmed for several special classes of semistandard lattices. A primary contribution of this thesis is a demonstration that the conjecture fails for many other classes of semistandard lattices.

Matthew Ross Gilliland, "Distributive Lattices and Weyl Characters of Exotic Type F4," defended April 2008.

ABSTRACT:
Posets and distributive lattices that model Weyl characters of the F4 type are investigated. For the two "smallest" characters, distributive lattice models and their posets of irreducibles obtained by Donnelly are presented. The existence of distributive lattice models for other "small" characters is explored here using known methods and some new approaches. One of the main contributions of this thesis is a demonstration that for certain small characters, distributive lattice models do not exist. Another contribution is the discovery of distributive lattice models for certain other small characters; these models were found using posets of irreducibles. Obtaining these new existence/nonexistience results was aided by a new concept presented here, the so-called "distributive core."