Recent Trends in Quasisymmetric Functions

Abstract

This article serves as an introduction to several recent developments in the study of quasisymmetric functions. The focus of this survey is on connections between quasisymmetric functions and the combinatorial Hopf algebra of noncommutative symmetric functions, appearances of quasisymmetric functions within the theory of Macdonald polynomials, and analogues of symmetric functions. Topics include the significance of quasisymmetric functions in representation theory (such as representations of the 0-Hecke algebra), recently discovered bases (including analogues of well-studied symmetric function bases), and applications to open problems in symmetric function theory.

Representation stability for sequences of 0-Hecke modules

“…H has concrete connections to QSym, the ring of quasisymmetric functions, as seen in Section 13. This ring has seen a resurgence of interest as of late, for a general survey we refer the reader to [25]. The connection between FI and the ring of symmetric functions is very explicit, in particular the authors in [37] prove that the Grothendieck group of FI-modules is isomorphic to two copies of the ring of symmetric functions.…”

Hopf Algebra Structure of Generalized Quasi-Symmetric Functions in Partially Commutative Variables

“…The theory of symmetric functions [72,52] is by now a well established subject with numerous applications in algebraic topology, combinatorics, representation theory, integrable systems and geometry. Quasi-symmetric functions, introduced by Gessel [33] (see also an earlier relevant work of Stanley [71]), are extensions of symmetric functions that are becoming of comparable importance [51,3,58]. As a graded Hopf algebra, the dual of the algebra of quasi-symmetric functions is the Hopf algebra of non-commutative symmetric functions introduced by Gelfand, Krob, Lascoux, Leclerc, Retakh and Thibon [32].…”