Hopf Algebras in Combinatorics

TL;DR

This paper surveys key Hopf algebras in combinatorics, covering their structures, identities, and applications in representation theory, with a focus on symmetric functions, quasisymmetric functions, and related algebraic objects.

Contribution

It provides a comprehensive, self-contained overview of important Hopf algebras in combinatorics, highlighting recent results and their applications.

Findings

Zelevinsky's characterization of symmetric functions as PSHs

Antipode formula for P-partition enumerators

Universal property of QSym by Aguiar-Bergeron-Sottile

Abstract

These notes -- originating from a one-semester class by their second author at the University of Minnesota -- survey some of the most important Hopf algebras appearing in combinatorics. After introducing coalgebras, bialgebras and Hopf algebras in general, we study the Hopf algebra of symmetric functions, including Zelevinsky's axiomatic characterization of it as a "positive self-adjoint Hopf algebra" and its application to the representation theory of symmetric and (briefly) finite general linear groups. The notes then continue with the quasisymmetric and the noncommutative symmetric functions, some Hopf algebras formed from graphs, posets and matroids, and the Malvenuto-Reutenauer Hopf algebra of permutations. Among the results surveyed are the Littlewood-Richardson rule and other symmetric function identities, Zelevinsky's structure theorem for PSHs, the antipode formula for P-partition enumerators, the Aguiar-Bergeron-Sottile universal property of QSym, the theory of Lyndon words, the Gessel-Reutenauer bijection, and Hazewinkel's polynomial freeness of QSym. The notes are written with a graduate student reader in mind, being mostly self-contained but requiring a good familiarity with multilinear algebra and -- for the representation-theory applications -- basic group representation theory.