UC Berkeley

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 $c_{\mu\nu}^{\lambda}$ appearing in the expression $s_{\mu}s_{\nu} = \sum_{\lambda} c_{\mu\nu}^{\lambda}s_{\lambda}$ 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.