Combinatorial Interpretation of the Kumar-Peterson Limit for sln(C) Demazure Characters and Gelfand Pattern Description of sln(C) Demazure Characters.

Given a semisimple Lie algebra, a dominant integral weight lambda, and a Weyl group element w, the Kumar-Peterson identity expresses the limit of a certain sequence of Demazure characters as a certain product over a subset of the positive roots. In Type A, the Demazure characters for a given n-partition lambda and a given permutation are also known as key polynomials. In this thesis we obtain a combinatorial interpretation of the Kumar-Peterson identity in Type A. Our combinatorial interpretation presents two product identities for the generating function for a set of certain reverse semistandard tableaux. The first right hand side is a product over the inversions of an inverse shuffle. The second right hand side is a product over the hooks of certain Hillman-Grassl boards. These identities can be viewed as combinatorial identities in and of themselves. We give bijective proofs of these identities which extend the Hillman-Grassl algorithm. We also give a Lie theoretic proof of the first identity by translating the Lie theoretic entities to combinatorial entities. The foremost special case of the second identity is equivalent to Gansner's identity for colored reverse plane partitions. That identity generalized identities found by Stanley, MacMahon, and Euler. Therefore this work places those identities in a Lie theoretic setting, and obtains the most general result of this kind within Type A. The foremost step in the Lie theoretic proof is to show that the limit of the Demazure polynomials can be found by calculating the direct limit of certain sets of Demazure tableaux. The principal tool we use in the Lie theoretic proof is Willis' scanning method for describing the Demazure tableaux. In the final chapter, we present a translation of this scanning method to Gelfand patterns.Doctor of Philosoph