Combinatorial models for representations of simple and affine Lie algebras

ABSTRACT
ACKNOWLEDGMENT
Introduction
A Combinatorial model for q-weight multiplicities for simple Lie algebras
Combinatorial Models for Representations of Affine Lie Algebras
An introduction to Kashiwara crystals
Crystals as combinatorial objects
Introduction to crystals of tableaux for type A
A Combinatorial model for q-weight multiplicities for simple Lie algebras
Background
Kostka-Foulkes Polynomials and the Kostant Partition Function
The Crystals B() and B^*()
The Main Result
A Crystal Structure on Kostant Partitions
The _i Operator on Kostant Partitions
An Involution on S_, and the Kostka-Foulkes Polynomials
Proof of Theorems 3.2.5 and 3.2.6
Proof of Theorems 3.2.7 and 3.2.8
Combinatorial Models for Representations of Affine Lie Algebras
Background
Root Systems
Kirillov-Reshetikhin (KR) crystals
The quantum alcove model
The bijection in types A_n-1 and C_n
The quantum alcove model and filling map in type A_n-1
The inverse map in type A_n-1
The quantum alcove model and filling map in type C_n
The inverse map in type C_n
The bijection in type B_n
The type B_n Kirillov-Reshetikhin crystals
The quantum alcove model and filling map in type B_n
The type B_n inverse map
Proof of Proposition 4.3.16
Necessary conditions for reordered columns
Necessary conditions for the construction of a segment of the quantum Bruhat path
Constructing a segment of the quantum Bruhat path.
Proof of Proposition 4.3.17
Classifying Split, Extended columns
Building a segment of the QBG Path between Split, Extended columns
The bijection in type D_n
Conclusion
Proving Conjectures 3.2.12 and 3.4.12
A statistic for Lusztig's q-analogue of weight multiplicities beyond type A
Extending the bijection to an explicit affine crystal isomorphism
A statistic for the energy function on affine crystals beyond type A