Summer School on Algebraic Combinatorics |
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Speaker: Nantel Bergeron (York University, Canada)
Title: Combinatorial Hopf Algebra, quasisymmetric functions and antipode formulas
Abstract: A combinatorial Hopf algebra is a graded-connected Hopf algebra with a given character. Using interesting examples, I will introduce these notions and what we can do with them.
For example take the Hopf algebra of equivalent classes of graphs with character $psi(G)=1$ if the graph is trivial and $0$ otherwise. Given such structure, we can construct a quasisymmetric function $Psi(G)$ that encode some combinatorial invariants of $G$. In particular, the chromatic polynomial $chi_G(x)$ of $G$. All these constructions preserve the Hopf structures, in particular one can see that
$$chi_G(-1) = psi(S(G)) $$
where $S$ is the antipode of the Hopf algebra of graphs. This antipode was explicitly computed by Humpert and Martin and involved acyclic orientation of $G$ and its quotients. From this one can recover a classical theorem of Stanley showing that $chi_G(-1)$ is the number of acyclic orientations of $G$. I will remain as elementary as possible to cover this and will assume only notion of linear algebras and tensor products.
Speaker: Sergey Fomin (University of Michigan, USA)
Title: Schur functions and Littlewood-Richardson Rules
Abstract: Schur functions form the most important basis in the ring of symmetric functions. We will review several definitions of the Schur functions, and discuss a general approach to obtaining Littlewood-Richardson-type rules that employs Schur functions in noncommuting variables.
Speaker: Ira Gessel (Brandeis University, USA)
Title 1: An Introduction to Symmetric Functions
Abstract 1: This talk will be an introduction to the theory of symmetric functions, including the important bases for symmetric functions, the scalar product, connections with representations of symmetric groups, and plethysm.
Title 2: An Introduction to Combinatorial Species
Abstract 2: The theory of combinatorial species is a method for counting labeled and unlabeled structures such as graphs of various types. In particular it provides a powerful alternative approach to problems traditionally solve by Pólya theory. I will discuss the basic definitions and examples, and describe the fundamental operations on species. I will also discuss the three fundamental generating functions associated with a species, in particular, the cycle index, which is a symmetric function associated with an action of the symmetric group.
Speaker: Jim Haglund (University of Pennsylvania, USA)
Title 1: The Combinatorics of Macdonald Polynomials
Abstract 1: Macdonald polynomials are symmetric functions in a set of variables X which also depend on a partition and two parameters q,t. Originally defined by Macdonald in terms of an orthogonality condition, in this talk we discuss a purely combinatorial formula for them due to Haiman, Loehr and the speaker. We show how to express Macdonald polynomials in terms of another family known as LLT polynomials. As a consequence the famous problem of finding a combinatorial expression for the coefficients in the Schur function of Macdonald polynomials is embedded in the corresponding problem of expanding LLT polynomials in the Schur basis. Some open bijective questions involving permutation statistics and Macdonald polynomials are also discussed.
Title 2: Diagonal harmonics and q,t-analogs of Catalan numbers and parking functions
Abstract 2: The combinatorics of Macdonald polynomials is closely connected to the combinatorics of the space of diagonal harmonics. We discuss the shuffle conjecture due to Haiman, Loehr, Remmel, Ulyanov and the speaker, which gives a combinatorial formula for the expansion of the character of diagonal harmonics into monomials, as a sum over weighted parking functions. This conjecture contains formulas giving q,t-analogs of Catalan numbers, parking functions, and other well-known combinatorial sequences. We mention that Carlsson and Mellit posted a preprint on the math arXiv last summer which seems to contain a proof of this conjecture.
Speaker: Alexander Yong (University of Illinois, USA)
Title: Kombinatorics
Abstract: I will survey an analogue of textbook tableau combinatorics (Young tableaux and Schur functions). This analogue originally arose to address questions about the K-theory of the Grassmannian. However, over the past decade, the combinatorics has become part of a broader conversation in algebraic and enumerative combinatorics, e.g., Hopf algebras, cyclic sieving, Demazure characters, homomesy, longest increasing subsequences of random words, poset edge densities and plane partitions. I will give an overview of results of many people including those I have jointly obtained with Oliver Pechenik (U. Illinois at Urbana-Champaign) and with Hugh Thomas (U. Quebec at Montreal).