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Operators on ktableaux and the kLittlewoodRichardson rule for a special case
 Iveson, Sarah Elizabeth
 Advisor(s): Haiman, Mark
Abstract
This thesis proves a special case of the $k$LittlewoodRichardson rule, which is analogous to the classical LittlewoodRichardson rule but is used in the case for $k$Schur functions. The classical LittlewoodRichardson 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 $k$LittlewoodRichardson coefficients that appear in the expansion of the product as a sum of $k$Schur functions. These $k$LittlewoodRichardson 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 $k$LittlewoodRichardson coefficients.
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