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Preprint
Dual immaculate creation operators and a dendriform algebra structure on the quasisymmetric functions
arXiv.org
30 Sep 2014
Abstract
Canad. J. Math. 69(2017), 21-53 (published version is version 5
without ancillary file) 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.
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Details
- Title
- Dual immaculate creation operators and a dendriform algebra structure on the quasisymmetric functions
- Creators
- Darij Grinberg
- Publication Details
- arXiv.org
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991021862368504721