Dual creation operators and a dendriform algebra structure on the quasisymmetric functions. (English) Zbl 1384.05158

Grinberg, Darij

Can. J. Math. 69, No. 1, 21-53 (2017).

Summary: The dual immaculate functions are a basis of the ring \(\mathrm{*QSym}\) 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 C. Berg et al. [Can. J. Math. 66, No. 3, 525–565 (2014; Zbl 1291.05206)], and have since been found to possess numerous nontrivial properties.
In this note, we prove a conjecture of M. 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 \(\mathrm{*QSym}\); we discuss the relation of this structure to known dendriform structures on the combinatorial Hopf algebras \(\mathrm{*FQSym}\) and \(\mathrm{*WQSym}\).

MSC:

05E05	Symmetric functions and generalizations
05E10	Combinatorial aspects of representation theory
16T05	Hopf algebras and their applications

Keywords:

combinatorial Hopf algebras; quasisymmetric functions; dendriform algebras; immaculate functions; Young tableaux