Operators on Compositions and Generalized Skew Pieri Rules
نویسنده
چکیده
Using operators on compositions we develop further both the theory of quasisymmetric Schur functions and of noncommutative Schur functions. By establishing relations between these operators, we show that the posets of compositions arising from the right and left Pieri rules for noncommutative Schur functions can each be endowed with both the structure of dual graded graphs and dual filtered graphs when paired with the poset of compositions arising from the Pieri rules for quasisymmetric Schur functions and its deformation. As a further application, we simplify the right Pieri rules for noncommutative Schur functions of Tewari. We then derive skew Pieri rules in the spirit of Assaf-McNamara for skew quasisymmetric Schur functions using the Hopf algebraic techniques of Lam-LauveSottile, and recover the original rules of Assaf-McNamara as a special case. Finally we apply these techniques a second time to obtain skew Pieri rules for skew noncommutative Schur functions.
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