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

Abstract

Abstract.e 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.e 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 de ned similarly to a semistandard Young tableau, but the shape is a composition rather than a partition, and only the rst 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:. , 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". e 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.

Noncommutative Bell polynomials and the dual immaculate basis

“…Therefore, Note 3.7. Theorem 3.6 provides an expression of the dual immaculate basis very similar to the one obtained by Grinberg [14]. However, our dendriform operations are different.…”

“…The formal sum of these "free Bell polynomials" satisfies a simple functional equation in terms of the dendriform structure of FQSym. This allows us to obtain expressions of the dual immaculate basis similar to (but different from) those of Grinberg [14]. Actually, Grinberg works directly at the level of quasi-symmetric functions, and his formula comes in fact from the tridendriform structure of WQSym.…”

Recent Trends in Quasisymmetric Functions

“…Grinberg recently proved Zabrocki's conjecture that the dual immaculate quasisymmetric functions can also be constructed using a variation on Bernstein's creation operators [47]. The dual immaculate quasisymmetric functions expand into positive sums of the monomial quasisymmetric functions, the fundamental quasisymmetric functions, and, recently shown in [3], the Young quasisymmetric Schur functions.…”

Shuffle-Compatible Permutation Statistics II: The Exterior Peak Set

“…We begin with some definitions. We will use some notations from [Grinbe16], but we set k = Q because we are working over the ring Q in this paper. Monomials always mean formal expressions of the form x a 1 1 x a 2 2 x a 3 3 · · · with a 1 + a 2 + a 3 + · · · < ∞ (see [Grinbe16, Section 2] for details).…”

“…Next, we shall recall the dendriform operations ≺ and on QSym studied in [Grinbe16], and we shall connect these operations back to LR-shuffle-compatibility. Since we consider this somewhat tangential to the present paper, we merely summarize the main results here; more can be found in [Grinbe18].…”