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Combinatorial Models For Representations Of Simple And Affine Lie Algebras, Adam Lee Schultze

Legacy Theses & Dissertations (2009 - 2024)

Part I: Koskta-Foulkes polynomials are Lusztig's q-analogues of weight multiplicities for irreducible representations of semisimple Lie algebras. It has long been known that these polynomials can be written with all non-negative coefficients. A statistic on semistandard Young tableaux with partition content, called \textit{charge}, was used to give a combinatorial formula exhibiting this fact in type $A$. Defining a charge statistic beyond type $A$ has been a long-standing problem. In the first part of this thesis, we take a completely new approach based on the definition of Kostka-Foulkes polynomials as an alternating sum over Kostant partitions, which can be thought of …