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/