Algebraic Combinatorics

Course Materials

• Possible Presentation Topics
• Josh Swanson's notes Done in real time!
• Hopf Algebras in Combinatorics by Darij Grinberg and Victor Reiner
• Hopf Algebras and Combinatorics online lectures by Federico Ardila with handwritten lecture notes available here and an outline of the online lectures is available here.
• Hopf Algebra Cheat Sheet by Kolya Malkin
• Hopfscotch: Our class blog. We are doing Ravi Vakil's 3 things game in public. Others are welcome to join the conversation or "ask us anything" in the upper lefthand corner of the blog.

Interesting Web Sites

• Combinatorics Seminar at UW
• Electronic Journal of Combinatorics
• Generatingfunctionology by Herb Wilf
• SAGE: Open Source Mathematical Software A collection of mathematical tools to do symbolic computation, combinatorial algorithms, graph manipulation, exact linear algebra, etc. Plus, its being developed right here at UW! Maple on the Web - Links to almost every Maple related web site.
• Sloane's On-Line Encyclopedia of Integer Sequences
• Wikipedia Dictionary of almost all terminology. This is a great teaching and learning tool.
• JSTOR Get papers from most AMS journals and other sources.
• Graphviz Package Graph drawing software. Here is my dot file for Bruhat order. (Note, I haven't tried it with the latest version of Graphviz yet.)

Syllabus

Summary:This three quarter topics course on Combinatorics includes Enumeration, Graph Theory, and Algebraic Combinatorics. Combinatorics has connections to all areas of mathematics and many other sciences including biology, physics, computer science, and chemistry. We have chosen core areas of study which should be relevant to a wide audience. The main distinction between this course and its undergraduate counterpart will be the pace and depth of coverage. In addition we will assume students have a basic knowledge of linear and abstract algebra. We will include many unsolved problems and directions for future research. The outline for each quarter is the following:

• Enumeration: Every discrete process leads to questions of existence, enumeration and optimization. This is the foundation of Combinatorics. In this quarter we will present the basic combinatorial objects and methods for counting various arrangements of these objects.
  • Basic counting methods.
  • Sets, multisets, permutations, and graphs.
  • Inclusion-exclusion.
  • Recurrence relations and integer sequences.
  • Generating functions.
  • Partially ordered sets.
  • Complexity Theory

• Random Graphs, Random Groups: Fifty years ago Erd\"os and Renyi began the study of random graphs. They answered questions such as how many randomly chosen edges does a graph on 100 vertices need to have before it is connected. The field they developed was not only able to answer probabilistic questions, but also questions about the existence of certain types of graphs. More recently Gromov started asking similar questions about what does a random group looks like. He conjectured (and Ollivier proved) that a random group with high probability is either hyperbolic or trivial.

  In this class we will look at these two closely related fields. We will touch on many branches of mathematics including graph theory, probability, topology and geometric group theory. While all of these are useful, none of them are prerequisites.

• Algebraic Combinatorics: The focus of this class will be on combinatorial Hopf algebras and diagram algebras. Diagram algebras/groups generalize the group algebra of the symmetric group where multiplication can be defined in term of concatenating string diagrams. Important examples include braid groups, the Braurer algebra, the Temperley-Lieb algebra, and topological quantum field theory on manifolds which appear in connection with knot theory. Combinatorial Hopf algebras have bases indexed by familiar objection in enumerative combinatorics. The primary example is the symmetric functions in a polynomial ring with $n$ variables. This course will survey recent work on symmetric functions, quasisymmetric functions, noncommuting symmetric functions as Hopf algebras and related representation theory including 0-Hecke algebra representations.

  There is no text but resources include Vic Reiner's notes on "Hopf Algebras In Combinatorics", Federico Ardila's lectures on "Hopf Algebras" and the book ``Coxeter Groups and Hopf Algebras'' by Marcelo Aguiar and Swapneel Mahajan.

  Course requirements: a strong background in algebra on par with our Math 504/5/6 and basic enumerative combinatorics on par with 461/462. No prior knowledge of Hopf algebras will be assumed.

• Fall: Enumerative Combinatorics; Volume 1 (second edition) by Richard Stanley, Cambridge Studies in Advanced Mathematics, 2011. The online version dated July 15, 2011 should be the same. Note the errata page.
• Winter: TBA.
• Spring: TBA

Additional Reading: A graduate level Combinatorics course is surprisingly close to the frontier of research in this area. This quarter every student will do presentations on research papers. Please look at the suggested papers or find some on your own. If you choose a paper on your own, be sure to discuss it with me asap.

Grading: The grade depends on the presentations and the discussion forum.

Tentative Schedule:

* Lecture 1: Introduction to Algebraic Combinatorics, diagram algebras, Hopf algebras, and the grand vision.

* Lecture 2: Representation theory of the symmetric groups and connections to symmetric functions.

* Lecture 3: Representation theory of the symmetric groups and connections to symmetric functions.

* Lecture 4: Representation theory of the symmetric groups and connections to symmetric functions.

* Lecture 5: Representation theory of the symmetric groups and connections to symmetric functions.

* Lecture 6: Representation theory of the symmetric groups and connections to symmetric functions.

* Lecture 7: Representation theory of the symmetric groups and connections to symmetric functions.

* Lecture 8: Hopf Algebras

* Lecture 9: Hopf Algebras

* Lecture 10: Hopf algebras

* Lecture 11: Quasisymmetric functions

* Lecture 12: Quasisymmetric functions

* Lecture 13: James Zhang guest lecture on Hopf algebras (April 28)

* Lecture 14: James Zhang guest lecture on Hopf algebras (April 30)

* Lecture 15: Quasisymmetric functions

* Lecture 16: (Monday May 5) Erik Slivken and Chris Fowler present "Hopf Algebras and Markov Chains" by Diaconis,Pang,Ram

* Lecture 17: (Wednesday May 7) Connor Ahlbach and Kolya Malkin present "A symmetric function generalization of the chromatic polynomial" by Stanley

* Lecture 18: (Friday May 9) Neil Goldberg presents "Coalgebras and bialgebras" by Joni and Rota

* Lecture 19: (Monday May 12) Hailun Zheng presents "On posets and Hopf algebras" by Ehrenborg

* Lecture 20: (Wednesday May 14) José Samper presents "Combinatorial Hopf algebras and generalized Dehn-Sommerville relations" by Aguiar,Bergeron, Sottile

* Lecture 21: (Friday May 16) Brendan Pawlowski presents "Partition algebras" by Halverson and Ram

* Lecture 22: (Monday May 19) Jair Taylor presents "Multipartite P-partitions and inner products of skew Schur functions" by Gessel

* Lecture 23: (Wednesday May 21) Michael Kiyo presents "A self paired Hopf algebra on double posets and a Littlewood-Richardson rule"

* Lecture 24: (Friday May 23) Open Problem Session

* Lecture 25: (Monday May 26) Memorial Day -- no class

* Lecture 26: (Wednesday May 28) Monty McGovern lectures

* Lecture 27: (Friday May 30) Macdonald polynomials, k-Schurs, and plethysm of Schur functions expand nicely into fundamental quasis.

* Lecture 28: (Monday June 2) 0-Hecke algebra

* Lecture 29: (Wednesday June 4) Edward Witten

* Lecture 30: (Friday June 6) Josh Swanson presents "Duality between quasisymmetric functions and the Solomon descent algebra" by Malvenuto and Reutenauer