MATH 280: Quasisymmetric and noncommutative symmetric functions
Fall 2012, UC Davis

Lectures: MW 5:10-6:30pm, MSB 3106
CRN 29278
Instructor: Anne Schilling, MSB 3222, phone: 554-2326,
Office hours: after class or by appointment
Text: The course will not strictly follow a particular texts. Some useful references are: We will also use recent papers which I will announce or hand out in class.
Computing: During class, I will illustrate some results using the open source computer algebra system Sage. When you follow the link, you can try it out yourself using Sage Online Notebook, or you can make an account on the LaCIM server. You can also run Sage on by typing the command `sage` to launch a Sage session in the terminal.
Grading: Most of all, this class is intended to expose graduate students to this active research area! We will discuss many open research problems. Registered student who would like to get experience with LaTeX are suggested to take notes for at least one class and hand them in in latex-format. A pdf version of the notes will be posted on the class web-site. You can work on this in pairs. The notes should be handed in no later than 6 days after the lecture. Please use the following template.
Anyone who will find mistakes or typos or has suggestions for improving the Primer on k-Schur functions will earn extra credit points.

Course description

Quasisymmetric functions are power series of bounded degree in variables x_1,x_2,... which are invariant under order-preserving shifts of the variables. This is to say that x_1^{m_1} ... x_k^{m_k} and x_{i_1}^{m_1} ... x_{i_k}^{m_k}$ have the same coefficient for any strictly increasing sequence of positive integers i_1< ... < i_k. These functions were defined in the 1980s by Ira Gessel, who developed many of their basic properties, and applied them to permutation enumerations.

In recent years, quasisymmetric functions have become increasingly important in several areas of mathematics, such as representation theory, algebraic combinatorics, and geometry. The subring QSym of quasisymmetric functions in the polynomial ring naturally contains the ring of symmetric functions, and is the dual Hopf algebra to the ring of symmetric functions in non-commutative variables (known as NSym). Quasisymmetric expansions have been used to prove the Schur-positivity of Macdonald polynomials and k-Schur functions, which can be interpreted as Schubert classes of the affine Grassmannian in geometry. They can also be considered as representatives of the characters for the representation theory of the 0-Hecke algebra. Recently, new bases for the space of quasisymmetric functions were defined which naturally refine the Schur basis for symmetric functions. Some of these bases are described by combinatorial objects known as composition-tableaux. Noncommutative symmetric functions in variables that for example satisfy plactic relations or the nilCoxeter algebra have also shown to be extremely useful tools in the combinatorics of affine Schubert calculus.

In this class we will introduce and explore quasisymmetric and noncommutative symmetric functions in these various settings.

Topics include:
Lecture notes: