François Bergeron.

• Algèbres de Hopf combinatoires (cas speciaux).
• Quotients d'algèbres de Hopf.
• Cas diagonal.

Aaron Lauve.

• Noncommutative Lagrange Inversion via Quasideterminants.

  I spent two days talking about quasideterminants and showed one application of questionable relevance to our course. There are many other applications of the quasideterminant (I hope to apply them to study the Descent Algebra in class). The one described here [PPR] is also of questionable relevance, but at least it's combinatorics.

• Idempotents in the Descent Algebra via Quasideterminants & NSYM.

  I expect I will say all of this myself, but if anybody is interested in stopping me and saying it, let me know soon. Browse [GKLLRT], Sections 3--5.

• The Other Hopf Algebra of `Noncommutative Symmetric Functions` "NCSYM".

  In class, I mentioned an important theorem of Wilson: Functions F which are polynomial in the y_k and symmetric in the x_i are in fact polynomial in the "a_k"---the coefficients of the minimal polynomial with roots {x₁, ... x_n}. That is, NSYM_n is finitely and freely generated by {a₁, ..., a_n} and NSYM may be truly viewed as `the algebra of noncommutative symmetric functions.' The proof is hard and shouldn't be used as a project.
  
  

  I also said in class, "if we consider functions F which are polynomial in the x_i and symmetric in the x_i then we find an algebra (NCSYM_n) which is neither finitely nor freely generated." HOWEVER, it is still an interesting algebra, studied by Wolf and Bergman-Cohn long ago, and more recently in [BRRZ,BZ]. It would be a super project to presentation this (Hopf) algebra. Proving that it is infinitely generated (for n ≤ ∞) would be a good goal.

• SSYM, an Alphabet Realization.

  The algebras SYM and QSYM (respectively, NCSYM and NCQSYM) may be viewed as the invariants of a group-action on Q[x] (inside Q⟨x⟩). The diagram François drew on the first day of class does not contain NCSYM and NCQSYM. Instead it contains NSYM and something he called SSYM [AS1]. There is a way to view NSYM and SSYM as the invariants of a group-action (same group, different action) on Q⟨x⟩. This starts with an alphabet realization of NSYM and SSYM, which I like very much (for one, it makes the comultiplication structure more natural). Presenting this alternate picture of NSYM and SSYM would make a nice project [DHT].

• Duals and Graded Duals of Hopf Algebras.

  One of the very nice features of the diagram François drew on the first day of class is that is `closed` under duality. I don't know if we are going to have a chance to speak about duals in class or not, but somebody should. Talk to any of us for more information.

• More Combinatorial Hopf Algebras.

  In this course, we are mostly dealing with Hopf algebras that look like Q[e₁, ...]. As I mentioned in class, there are some much more interesting Hopf algebras out there. Grossman-Larsen and Connes-Kreimer come to mind [AS3]. If you would like to tell us about your favorite combinatorial Hopf algebra, I'm sure we'd be delighted to listen (see also [AS2,NT,E]).

• Schur functions in SYM.

  We have mentioned Schur functions and skew-Schur functions, but never really explained why they are symmetric (i.e. why they belong to SYM). This paper does this and more, and is just the right length for a presentation: [Stm] A Concise Proof of the Littlewood-Richardson Rule.

Franco Saliola.

• Coxeter Groups

  There are many topics one can choose here. A presentation on the definition and general properties of Coxeter groups would fill a class. A discussion of the geometric ideas behind Coxeter groups is enough to fill a class. A proof on the classification of Coxeter groups can be presented in one class. Someone can discuss the combinatorics of Coxeter groups. See the section on Coxeter groups in the references below.

• Hyperplane Arrangements

  An introduction to the study of hyperplane arrangements, which is different from what I presented in class. See Richard Stanley's notes. Someone can present the theory, examples, and results from the theory of random walks on the chambers of a hyperplane arrangement.

• Poset Topology.

  I think this is a really beautiful subject, and there are many directions that this project can take. You can study group representations afforded by the (co)homology of a poset. You can discuss the homotopy types of posets. You can talk about shellability, edge labelling techniques, etc. See the notes by Michelle Wachs.

• Combinatorial Algebra

  Combinatorial Hopf algebras; Combinatorial Representation Theory; Quivers; Spectral Sequences in Combinatorics, Hochschild (co)homology of the incidence algebra of a poset. See the papers in the references under Combinatorial Algebra.

• [BBHT] F Bergeron, N Bergeron, Howlett, Taylor. A Decomposition Of The Descent Algebra Of A Finite Coxeter Group.
• [BB] F Bergeron, N Bergeron, A Decomposition of the Descent Algebra of the Hyperoctahedral Group I.
• [GR] A Garsia, C Reutenauer. A Decomposition Of Solomons Descent Algebra.
• [H] C Hohlweg. Properties Of The Solomon Algebra Homomorphism.
• [Sc1] M Schocker. Descent Algebra Of The Symmetric Group.
• [Sc2] M Schocker. Peak Algebra Revisted. (The peak algebra is a subalgebra of the descent algebra.)
• [So] L Solomon. A Mackey Formula In The Group Ring Of A Coxeter Group. (The introduction of the descent algebra.)
• [T] J Tits. Appendix: Two Properties Of Coxeter Complexes. (Appendix to the above.)

• [At] C A Athanasiadis. Deformations of Coxeter hyperplane arrangements and their characteristic polynomials.
• [BaD] D Bayer, P Diaconis. Trailing The Dovetail Shuffle To Its Lair.
• [BHR] P Bidigare, P Hanlon, D Rockmore. A Combinatorial Description Of The Spectrum For The Tsetlin Library and Generalizations to Hyperplane Arrangements.
• [BBD] L Billera, K Brown, P Diaconis. Random Walks And Plane Arrangements In Three Dimensions.
• [B1] K Brown. Rings, Semigroups and Markov Chains.
• [B2] K Brown. Semigroup And Ring Theoretical Methods In Probability.
• [BrD] K Brown, P Diaconis. Random Walks And Hyperplane Arrangements.
• [D] P Diaconis. Mathematical Developments From The Analysis Of Riffle Shuffling.
• [F] J Fulman. Descent Algebras, Hyperplane Arrangements And Shuffling Cards.
• [Sa] B Sagan. Why the Characteristic Polynomial Factors.
• [St1] R Stanley. An Introduction To Hyperplane Arrangements.

• [Al] D Allcock. Classification Of Finite Reflection Groups.
• [B3] K Brown. What Is A Building?
• [Ha] P de la Harpe. Invitation To Coxeter Groups.
• [Ho] B Howlett. Introduction to Coxeter Groups.
• [K] D Krammer. The Conjugacy Problem for Coxeter Groups.

• [ABB] J C Aval, F Bergeron, N Bergeron, Diagonal Temperley-Lieb Invariants and Harmonics.
• [AM] M Aguiar, S Mahajan. Coxeter Groups And Hopf Algebras I.
• [BR] H Barcelo, A Ram. Combinatorial Representation Theory.
• [Be] F Bergeron. Algèbre et combinatoire. (Anciennes notes de cours)
• [BGP] I N Bernstein, I M Gel'fand, V A Ponomarev. Coxeter Functors And Gabriel's Theorem.
• [C] P Cartier. A Primer in Hopf Algebras. (Includes a chapter on applications to combinatorics.)
• [Ch] T Chow. You Could Have Invented Spectral Sequences. (With the following.)
• [Conf] Various authors. Combinatorial Commutative Algebra.
• [K] D Kozlov. Spectral Sequences On Combinatorial Simplicial Complexes. (See above.)
• [Sk] E Skoldberg. Going From Cohomology To Hochschild Cohomology.

• [St2] R Stanley Some Aspects Of Groups Acting On Finite Posets.
• [W] M Wachs Poset Topology Tools And Applications.

• [BBRZ] N Bergeron, C Reutenauer, M Rosas, M Zabrocki Noncommutative Invariants and Coinvariants of the Symmetric Group (the other algebra of "noncommutative symmetric functions")
• [BZ] N Bergeron, M Zabrocki NCSYM and NCQSYM are Free and Cofree (more about the previous item)
• [GGRW] I M Gelfand, S Gelfand, V Retakh, R Wilson Quasideterminants (a survey paper)
• [GKLLRT] I M Gelfand et al. (the "hexagon paper") Noncommutative Symmetric Functions (abstractly)
• [O] B Osofsky Right Roots of Polynomials (Noncommutative symmetric functions… for real!)
• [PPR] A Postnikov, I Pak, V Retakh Noncommutative Lagrange Theorem and Inversion Polynomials (via quasideterminants!)

• [AS1] M Aguiar, F Sotille Structure of SSYM (the Malvenuto-Reutenauer Hopf algebra of Permutations)
• [AS2] M Aguiar, F Sotille The Loday-Ronco Hopf Algebra (of rooted planar binary trees)
• [AS3] M Aguiar, F Sotille Cocommutative Hopf algebras of Permutations and Trees
• [DHT] G Duchamp, F Hivert, J-Y Thibon NSF VI ("free quasi-symmetric functions" and related algebras)
• [E] R Ehrenborg On Posets and Hopf Algebras (guess)
• [Hi1] F Hivert Hecke Algebras,… Quasi-Symmetric Functions (Quasi-Symmetric functions are "symmetric")
• [Hi2] F Hivert Local Action… Quasi-Symmetric Functions ("r-QuasiSymmetric functions")
• [NT] J-C Novelli, J-Y Thibon Parking Functions and Descent Algebras (more Hopf algebras on words)
• [Stm] J Stembridge Concise Proof of the Littlewood-Richardson Rule (just what it says)