Symmetric functions is a lively subject developing rapidly for the last 200

years, at the crossroads of combinatorics and representation theory. We will

start from scratch (elementary symmetric polynomials) and hopefully get to

Macdonald polynomials. Everybody starting from the 2nd year students is

welcome, but some basic representation theory will be helpful.

 

Prerequisites: basic linear and multilinear algebra (tensor products, multilinear maps). 

Basic group theory and basic representation theory.

 

Curriculum:

Partitions.

Symmetric functions ring.

Schur functions.

Orthogonality.

Skew Schur functions.

Transition matrices.

Characters of symmetric groups.

Plethysm

Littlewood-Richardson rule

Polynomial functors

Hall algebra

Hall polynomials

Hall-Littlewood functions.

Green functions.

Parabolic induction.

Characters of general linear groups over finite fields.

Macdonald polynomials.

 

Textbooks:

I.G.Macdonald, Symmetric Functions and Hall Polynomials, second edition.

Oxford University Press, 1995.

A.Postnikov https://math.mit.edu/~apost/courses/18.218/