Hopf Algebras in Combinatorics

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.

282 pages. The version on my website (http://www.cip.ifi.lmu.de/~grinberg/algebra/HopfComb.pdf) is likely to be updated more frequently. Solutions to the exercises can be found in http://www.cip.ifi.lmu.de/~grinberg/algebra/HopfComb-sols.pdf , or as an ancillary file here. Version v7 has more details, a few more exercises and fewer errors. Comments are welcome!

References in corpus (5)

• Hopf algebras, from basics to applications to renormalization
• The characteristic polynomial of the Adams operators on graded connected Hopf algebras
• Kronecker coefficients for one hook shape
• A historical survey of P-partitions
• Sequences and nets in topology

Cited by in corpus (56)

• Hopf Algebras and Markov Chains
• Involution words: counting problems and connections to Schubert calculus for symmetric orbit closures
• Order Quasisymmetric Functions Distinguish Rooted Trees
• Double posets and the antipode of QSym
• Notes on the combinatorial fundamentals of algebra
• Petrie symmetric functions
• The Hopf monoid of hypergraphs and its sub-monoids: basic invariant and reciprocity theorem
• Ordered set partitions and the 0-Hecke algebra
• GL_n(F_q)-analogues of factorization problems in the symmetric group
• Enriched set-valued P-partitions and shifted stable Grothendieck polynomials
• A tableau approach to the representation theory of 0-Hecke algebras
• The Pelletier-Ressayre hidden symmetry for Littlewood-Richardson coefficients
• Cancelation free formula for the antipode of linearized Hopf monoid
• Quasisymmetric functions for nestohedra
• Hopf algebra structure of generalized quasi-symmetric functions in partially commutative variables
• Counting faces of nestohedra
• Linear compactness and combinatorial bialgebras
• The decomposition of 0-Hecke modules associated to quasisymmetric Schur functions
• Markov Chains from Descent Operators on Combinatorial Hopf Algebras
• A cancellation-free antipode formula (for uniform matroids) for the restriction-contraction matroid Hopf algebra
• Combinatorial reciprocity for non-intersecting paths
• Universality of swap for qudits: a representation theory approach
• Card-Shuffling via Convolutions of Projections on Combinatorial Hopf Algebras
• A lifting of the Goulden-Jackson cluster method to the Malvenuto-Reutenauer algebra
• Representation stability for sequences of 0-Hecke modules
• Matroid-minor Hopf algebra: a cancellation-free antipode formula and other applications of sign-reversing involutions
• Rigidity for the Hopf algebra of quasi-symmetric functions
• An Hopf algebra for counting simple cycles
• Signatures of graphs for bicommutative Hopf algebras
• Cylindric Young Tableaux and their Properties
• On the Combinatorics of the Universal Enveloping Algebra Uh(sl2)
• Theta maps for combinatorial Hopf algebras
• Extended Schur functions and bases related by involutions
• Automorphisms on the ring of symmetric functions and stable and dual stable Grothendieck polynomials
• Weighted quasisymmetric enumerator for generalized permutohedra
• Shifted combinatorial Hopf algebras from $K$-theory
• Decompositions of Nonlinear Input-Output Systems to Zero the Output
• Bialgebras for Stanley symmetric functions
• The enriched $q$-monomial basis of the quasisymmetric functions
• Chow rings and augmented Chow rings of uniform matroids and their $q$-analogs
• Bases for Quotients of Symmetric Polynomials
• On the Hopf algebra of multi-complexes
• Weighted $\mathsf{P}-$partitions enumerator
• Three variations on the linear independence of grouplikes in a coalgebra
• Hecke algebras of simply-laced type with independent parameters
• $P$-Partitions and Quasisymmetric Power Sums
• Compositions with restricted parts
• A uniform generalization of some combinatorial Hopf algebras
• On the square of the antipode in a connected filtered Hopf algebra
• Algorithmic and algebraic aspects of unshuffling permutations
• (Weak) incidence bialgebras of monoidal categories
• Supercharacters of unipotent and solvable groups
• Positive Self-Dual Hopf Algebras of Galois Characters
• A generalization of the dual immaculate quasisymmetric functions in partially commutative variables
• Hopf algebras from poset growth models
• The Bernstein homomorphism via Aguiar-Bergeron-Sottile universality