Abstract
We define a $0$-Hecke action on composition tableaux, and then use it to derive $0$-Hecke modules whose quasisymmetric characteristic is a quasisymmetric Schur function. We then relate the modules to the weak Bruhat order and use them to derive a new basis for quasisymmetric functions. We also classify those modules that are tableau-cyclic and likewise indecomposable. Finally, we develop a restriction rule that reflects the coproduct of quasisymmetric Schur functions. Nous définissons une action $0$-Hecke sur les tableaux de composition, et ensuite nous l’utilisons pour dériver les modules $0$-Hecke dont la caractéristique quasi-symétrique est une fonction de Schur quasi-symétrique. Nous mettons les modules en relation avec l’ordre de Bruhat faible et les utilisons pour dériver une nouvelle base pour les fonctions quasi-symétriques. Nous classons aussi ces modules qui sont tableau-cycliques et aussi indécomposable. Enfin, nous développons une règle de restriction qui reflète le coproduit des fonctions de Schur quasi-symétriques.
Highlights
Quasisymmetric functions were first defined by Gessel [Ges84] in the 1980s as weight enumerators for labelled posets
The Hopf algebra of quasisymmetric functions, QSym, has arisen in a variety of contexts including the study of Lie representations, riffle shuffles, random walks, and the representation theory of Hecke algebras
QSym contains the Hopf algebra of symmetric functions, Sym, as a subalgebra, which is a central object of study in algebraic combinatorics due in no small part to its basis of Schur functions that arise, for example, in the representation theory of the symmetric group, as generating functions for Young tableaux, and in Schubert calculus
Summary
Quasisymmetric functions were first defined by Gessel [Ges84] in the 1980s as weight enumerators for labelled posets. A new basis for QSym was discovered, which arises through the combinatorics of Macdonald polynomials and reflects many properties of Schur functions. These functions were called quasisymmetric Schur functions and properties reflected include that they yield quasisymmetric Kostka numbers; exhibit quasisymmetric Pieri rules and a quasisymmetric Littlewood-Richarsdon rule. Tewari and Stephanie J. van Willigenburg to resolving the conjecture that QSym over Sym has a stable basis, and have motivated the search for other Schur-like bases of QSym such as row-strict quasisymmetric functions These details and more on quasisymmetric Schur functions, including full citations for all of the above, can be found in [LMvW13]. Due to space constraints we will not include any proofs, we will often very briefly indicate the proof techniques involved, for the interested reader
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