Math 740 - Symmetric Functions (Spring 2017)

Lecture times and location: TR 9:30AM - 10:45AM, Van Vleck B325
Textbooks:
- I. G. Macdonald, Symmetric Functions and Hall Polynomials, second edition
- Richard Stanley, Enumerative Combinatorics Volume 2 (Chapter 7)
Macdonald's text is much more comprehensive, so is a useful reference to have, especially for later topics. For the basic theory, I find Chapter 7 of Stanley's book to be much easier for a first read, so we'll follow it in the beginning.
Course website: http://math.wisc.edu/~svs/740/
Instructor: Steven Sam email
Instructor office hours: 321 Van Vleck, by appointment

Notes for course (updated 4/27/17) — errata for notes by Darij Grinberg
Homework (updated 4/14/17)

Schedule

Below, section numbers refer to Stanley's book. We're not currently using it though.


Jan 17	What are symmetric functions? 7.1 Polynomial representations of general linear groups 7.A2 Partitions 7.2
Jan 19	Monomial symmetric functions 7.3 Elementary symmetric functions 7.4
Jan 24	The involution ω 7.6 Complete homogeneous symmetric functions 7.5 Power sum symmetric functions 7.7
Jan 26	Scalar product 7.9 Some representation theory Semistandard Young tableaux 7.10

Jan 31	Schur functions 7.10 RSK algorithm 7.11
Feb 2	RSK algorithm and corollaries 7.12-7.14 Classical definition of Schur functions 7.15

Feb 7	Jacobi-Trudi identity 7.16 Summary of representation theory of finite groups
Feb 9	Representations of the symmetric group 7.18

Feb 14	Murnaghan-Nakayama rule 7.17
Feb 16	Schur-Weyl duality Grassmannians and Schubert decomposition

Feb 21	Schubert varieties Review of cohomology ring
Feb 23	Schubert calculus

Feb 28	Chern roots/classes Lines on a cubic surface
Mar 2	Formulas for number of SYT and SSYT

Mar 7	Littlewood-Richardson coefficients Combinatorics of finite abelian p-groups
Mar 9	Hall algebras and Hall polynomials

Mar 14	Hall-Littlewood symmetric functions, definitions
Mar 16	Hall-Littlewood symmetric functions, Pieri rule Connection to Hall polynomials

Mar 21	Spring break - no class
Mar 23	Spring break - no class

Mar 28	Hall-Littlewood function identities
Mar 30	t-deformation of Jacobi-Trudi identity t-deformation of inner product

Apr 4	Schur Q-functions
Apr 6	Projective representation theory

Apr 11	Clifford algebras
Apr 13	Guest lecture: Daniel Erman

Apr 18	no class
Apr 20	The basic spin representation

Apr 25	Frobenius characteristic map for projective representations
Apr 27	Schur Q-functions: wrapup

May 2	no class
May 4	Guest lecture: Jordan Ellenberg