UBC Theses and Dissertations
Operators on compositions and noncommutative Schur functions Tewari, Vasu
In this thesis, we study a natural noncommutative lift of the ubiquitous Schur functions, called noncommutative Schur functions. These functions were introduced by Bessenrodt, Luoto and van Willigenburg and resemble Schur functions in many regards. We prove some new results for noncommutative Schur functions that are analogues of classical results, and demonstrate that the resulting combinatorics in this setting is equally rich. First we prove a Murnaghan-Nakayama rule for noncommutative Schur functions. In other words, we give an explicit combinatorial formula for expanding the product of a noncommutative power sum symmetric function and a noncommutative Schur function in terms of noncommutative Schur functions. In direct analogy to the classical Murnaghan-Nakayama rule, the summands are computed using a noncommutative analogue of border strips, and have coefficients ±1 determined by the height of these border strips. The rule is proved by interpreting the noncommutative Pieri rules for noncommutative Schur functions in terms of box adding operators on compositions. We proceed to give a backward jeu de taquin slide analogue on semistandard reverse composition tableaux. These tableaux were first studied by Haglund, Luoto, Mason and van Willigenburg when defining quasisymmetric Schur functions. Our algorithm for performing backward jeu de taquin slides on semistandard reverse composition tableaux results in a natural operator on compositions that we call the jdt operator. This operator in turn gives rise to a new poset structure on compositions whose maximal chains we enumerate. As an application, we also give new right Pieri rules for noncommutative Schur functions that use the jdt operators, in contrast to the left Pieri rules given by Bessenrodt, Luoto and van Willigenburg.
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