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Dual Creation Operators and a Dendriform Algebra Structure on the Quasisymmetric Functions

  Published:2016-07-12
 Printed: Feb 2017
  • Darij Grinberg,
    Mathematics Department, Massachusetts Institute of Technology, Cambridge, MA
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Abstract

The dual immaculate functions are a basis of the ring $\operatorname*{QSym}$ of quasisymmetric functions, and form one of the most natural analogues of the Schur functions. The dual immaculate function corresponding to a composition is a weighted generating function for immaculate tableaux in the same way as a Schur function is for semistandard Young tableaux; an " immaculate tableau" is defined similarly to be a semistandard Young tableau, but the shape is a composition rather than a partition, and only the first column is required to strictly increase (whereas the other columns can be arbitrary; but each row has to weakly increase). Dual immaculate functions have been introduced by Berg, Bergeron, Saliola, Serrano and Zabrocki in arXiv:1208.5191, and have since been found to possess numerous nontrivial properties. In this note, we prove a conjecture of Mike Zabrocki which provides an alternative construction for the dual immaculate functions in terms of certain "vertex operators". The proof uses a dendriform structure on the ring $\operatorname*{QSym}$; we discuss the relation of this structure to known dendriform structures on the combinatorial Hopf algebras $\operatorname*{FQSym}$ and $\operatorname*{WQSym}$.
Keywords: combinatorial Hopf algebras, quasisymmetric functions, dendriform algebras, immaculate functions, Young tableaux combinatorial Hopf algebras, quasisymmetric functions, dendriform algebras, immaculate functions, Young tableaux
MSC Classifications: 05E05 show english descriptions Symmetric functions and generalizations 05E05 - Symmetric functions and generalizations
 

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