Dual immaculate creation operators and a dendriform algebra structure on the quasisymmetric functions
arXiv:1410.0079 · doi:10.4153/CJM-2016-018-8
Abstract
The dual immaculate functions are a basis of the ring 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 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 QSym; we discuss the relation of this structure to known dendriform structures on the combinatorial Hopf algebras FQSym and WQSym.
41 pages. Updated postprint (version 5 was published in Canad. J. of Math.; published version is version 5 without ancillary file). The ancillary PDF file (compiled from the same source) has more details and proves Proposition 5.7. Comments are welcome! Version 10 adds missing assumptions to Theorem 3.7 and Lemma 5.4. Includes tweaked allrunes package to work around LaTeX incompatibility
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- The Hopf algebras of symmetric functions and quasisymmetric functions in non-commutative variables are free and cofree
- Dendriform Equations
- Indecomposable modules for the dual immaculate basis of quasi-symmetric functions