TY - JOUR TI - Two problems in representation theory: affine Lie algebras and algebraic combinatorics DO - https://doi.org/doi:10.7282/t3-d7ga-4c38 PY - 2019 AB - In this dissertation, we investigate two topics with roots in representation theory. The first topic is about twisted affine Kac-Moody algebras and vector spaces spanned bytheir characters. Specifically, the space spanned by the characters of twisted affine Lie algebras admit the action of certain congruence subgroups of SL (2, Z). By embedding the characters in the space spanned by theta functions, we study an SL (2, Z)-closure of the space of characters. Analogous to the untwisted affine Lie algebra case, we construct a commutative associative algebra (fusion algebra) structure on this space through the use of the Verlinde formula and study important quotients. Unlike the untwisted cases, some of these algebras and their quotients, which relate to the trace of diagram automorphisms on conformal blocks, have negative structure constants with respect to the (usual) basis indexed by the dominant integral weights of the Lie algebra. We give positivity conjectures for the new structure constants and prove them in some illuminating cases. We then compute formulas for the action of congruence subgroups on these character spaces and give explicit descriptions of the quotients using the affine Weyl group. The second topic concerns algebraic combinatorics and symmetric functions. In statistics, zonal polynomials and Schur functions appear when taking integrals over certain compact Lie groups with respect to their associated Haar measures. Recently, a conjecture, related to certain integrals of statistical interest, was proposed by D. Richards and S. Sahi. This conjecture asserts that certain linear combinations of Jack polynomials, a one-parameter family of symmetric polynomials that generalizes the zonal and Schur polynomials, are non-negative when evaluated over a certain cone. In the second part of this dissertation, we investigate these conjectures for Schur polynomials and give a refined version of the conjecture. In addition, we prove some cases and arrive at certain seemingly new combinatorial results. In an important instance, we give an analogous result for Jack polynomials. KW - Affine Lie algebras KW - Mathematics KW - Lie algebras LA - English ER -