Math 8680: Topics in Combinatorics
Hopf algebras in combinatorics

Fall 2012

Prerequisites: The equivalent of graduate algebra, including multilinear algebra, e.g. tensor products. Familiarity with representation theory of finite groups would be helpful, and some knowledge of symmetric functions is helpful, but not essential. (No prior knowledge of Hopf algebras will be assumed.)  
Instructor: Victor Reiner (You can call me "Vic"). 
Office: Vincent Hall 256
Telephone (with voice mail): 625-6682
Classes: Monday and Wednesdays 4:00 - 5:30pm in Vincent Hall 206 
Office hours: Monday and Wednesdays 3:30-4:00, and by appointment. 
Course content: Certain Hopf algebras arise in combinatorics because they have bases naturally parametrized by combinatorial objects (partitions, compositions, permutations, tableaux, binary trees, etc). The rigidity in the structure of a Hopf algebra sometimes leads to simpler proofs and explanations, and many interesting invariants of combinatorial objects turn out to be evaluations of Hopf morphisms or characters on Hopf algebras.
This course will focus on examples....
  • the standard ones: tensor, symmetric, divided power, exterior algebras, and universal enveloping algebras,
  • review of the ring of symmetric functions,
    • emphasizing its Hopf structure (coalgebra, antipode),
    • its use in the Lam(-Lauve-Sottile) proof of the Assaf-McNamara skew Pieri rule,
    • Zelevinsky's theorem: the ring of symmetric functions is the unique indecomposable Hopf algebra with a self-dual basis and positive structure constants,
    • its role in describing complex characters of
      • the symmetric group,
      • wreath products of the symmetric group with a finite group,
      • the finite general linear groups, and
      • the Hall algebra
  • quasisymmetric functions, including
    • duality with noncommutative symmetric functions,
    • P-partitions,
    • interpretation in terms of representations of 0-Hecke algebras (via representation theory of J-trivial monoids)
    • Aguiar-Bergeron-Sottile character theory
    • Hoffman's character and multiple zeta values
  • incidence algebras, Eulerian poset and polyotope Hopf algebras, and the cd-index.
  • the Malvenuto-Reutenauer Hopf algebra of permutations?
  • Hopf algebras of graphs and the Stanley chromatic symmetric function?
  • the Crapo-Schmitt matroid Hopf algebra and associated quasisymmetric function?
  • Schur's Q-function algebra inside symmetric functions, and Stembridge's peak subalgebra of quasisymmetric functions?
Grading: Grad students who are registered for the class and want to get an A should attend regularly, and give a talk once during the semester on some topic related to Hopf algebras. Either pick a paper from the resource list below, or come up with a suggestion on your own, and convince me.
Texts: There is no required text, but here is the latest version of my course notes, partly filled-in, and partly skeleton, and now co-authored with Darij Grinberg.
Here is the
arXiv version. Typos, suggested edits, additions, deletions, are welcome.
Some sources that I will consult a lot are
  • A.V. Zelevinsky,
    Representations of finite classical groups. A Hopf algebra approach.
    Lecture Notes in Mathematics, 869. Springer-Verlag, Berlin-New York, 1981.
  • S. Montgomery,
    Hopf algebras and their actions on rings.
    CBMS Regional Conference Series in Mathematics, 82. Amer. Math. Soc., Providence, RI, 1993.
More resources:
  • Federico Ardila's Spring 2012 Hopf algebra course at SFSU and Universidad De Los Andes
  • Spencer Bloch's Fall 2007 Hopf algebra course at U. Chicago.
  • Jean-Yves Thibon's papers page
  • Marcelo Aguiar's home page
  • Richard Ehrenborg's home page
  • M. Aguiar and F. Sottile. Structure of the Malvenuto-Reutenauer Hopf algebra of permutations
  • M. Aguiar, C. Andre + 26 other authors. Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras
  • M. Aguiar and F. Ardila. The Hopf monoid of generalized permutahedra.
  • M. Aguiar, N. Bergeron, and F. Sottile. Combinatorial Hopf algebras and generalized Dehn-Sommerville relations.
  • A. Baker and B. Richter, Quasisymmetric functions from a topological point of view
  • N. Bergeron and M. Zabrocki. The Hopf algebras of symmetric functions and quasisymmetric functions in non-commutative variables are free and cofree
  • L.J. Billera, Flag Enumeration in Polytopes, Eulerian Partially Ordered Sets and Coxeter Groups
  • P. Cartier. A primer on Hopf algebras.
  • T. Denton, F. Hivert, A. Schilling, and N. Thiery. On the representation theory of finite J-trivial monoids.
  • P. Diaconis, A. Pang and A. Ram. Hopf algebras and Markov chains: Two examples and a theory
  • G. Duchamp, P. Blasiak, A. Horzela, K. Penson, A. Solomon. Hopf algebras in general and in combinatorial physics: a practical introduction.
  • R. Ehrenborg. On posets and Hopf algebras.
  • A.E. Ellis and M. Khovanov. The Hopf algebra of odd symmetric functions
  • S. Forcey, A. Lauve, and F. Sottile. New Hopf structures on binary trees.
  • R. Henderson, The Algebra Of Multiple Zeta Values
  • F. Hivert. An introduction to combinatorial Hopf algebras: examples and realizations
  • F. Hivert, J.-C. Novelli and J.-Y. Thibon. Commutative combinatorial Hopf algebras.
  • M.E. Hoffman, Combinatorics of rooted trees and Hopf algebras
  • M.E. Hoffman. A character on the quasi-symmetric functions coming from multiple zeta values
  • S. Joni and G.-C. Rota. Coalgebras and bialgebras in combinatorics.
  • T. Lam and P. Pylyavskyy, Combinatorial Hopf algebras and K-homology of Grassmanians.
  • A. Lauve, T. Lam, F. Sottile. Skew Littlewood-Richardson rules from Hopf Algebras.
  • J.-L. Loday and M. Ronco. Combinatorial Hopf algebras.
  • J.-L. Loday and M. Ronco. Hopf algebra of the planar binary trees.
  • C. Malvenuto and C. Reutenauer. Duality between quasi-symmetric functions and the Solomon descent algebra.
  • J. Milnor and J. Moore. On the structure of Hopf algebras.
  • N. Reading. Lattice congruences, fans and Hopf algebras.
  • O. Schiffmann. Lectures on Hall algebras.
  • W. Schmitt. Incidence Hopf algebras.
  • J.-Y. Thibon. An Introduction to Noncommutative Symmetric Functions.
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