Operators on k-tableaux and the k -Littlewood--Richardson rule for a special case

This thesis proves a special case of the k-Littlewood--Richardson rule, which is analogous to the classical Littlewood--Richardson rule but is used in the case for k-Schur functions. The classical Littlewood--Richardson rule gives a combinatorial formula for the coefficients clmn appearing in the expression smusnu = lcl mnsl for two Schur functions multiplied together. k-Schur functions are another class of symmetric functions which were introduced by Lapointe, Lascoux, and Morse and are indexed by and related to k-bounded partitions. We investigate what occurs when multiplying two k-Schur functions with some restrictions. More specifically, we investigate what happens when a k-Schur function is multiplied by a k-Schur function corresponding to a partition of length two. In this restricted case we are able to provide a combinatorial description of the k-Littlewood--Richardson coefficients that appear in the expansion of the product as a sum of k-Schur functions. These k-Littlewood--Richardson coefficients can be computed in terms of the number of k-tableaux with a certain property we call k-lattice. Furthermore, we conjecture that the result holds for any k-Schur functions, even when no restrictions are imposed. The proofs presented rely on a class of operators on k-tableaux which we introduce that are similar to the crystal operators on classical tableaux, but we provide a specific example that implies they are not actually crystal operators on k -tableaux. In addition to this, we also provide numerous examples and dedicate a chapter to examples of computation for some k-Littlewood--Richardson coefficients.