The enriched \(q\)-monomial basis of the quasisymmetric functions

Darij Grinberg¹ ; Ekaterina Vassilieva²

¹University of Minnesota, Twin Cities
²École Polytechnique

The electronic journal of combinatorics, Tome 31 (2024) no. 4

Résumé

We construct a new family $\left( \eta_{\alpha}^{\left( q\right) }\right)_{\alpha\in\operatorname*{Comp}}$ of quasisymmetric functions for each element $q$ of the base ring. We call them the "enriched $q$-monomial quasisymmetric functions". When $r:=q+1$ is invertible, this family is a basis of $\operatorname*{QSym}$. It generalizes Hoffman's "essential quasi-symmetric functions" (obtained for $q=0$) and Hsiao's "monomial peak functions" (obtained for $q=1$), but also includes the monomial quasisymmetric functions as a limiting case. We describe these functions $\eta_{\alpha}^{\left( q\right) }$ by several formulas, and compute their products, coproducts and antipodes. The product expansion is given by an exotic variant of the shuffle product which we callthe "stufufuffle product'' due to its ability to pick several consecutive entries from each composition. This "stufufuffle product'' has previously appeared in recent work by Bouillot, Novelli and Thibon, generalizing the "block shuffle product'' from the theory of multizeta values.

Détail

DOI : 10.37236/12409

Classification : 05E05, 05A30, 11M32
Mots-clés : enriched \(q\)-monomial quasisymmetric functions, monomial peak functions, stufufuffle product