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)
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 21Spring break - no class
Mar 23Spring break - no class
 
Mar 28Hall-Littlewood function identities
Mar 30t-deformation of Jacobi-Trudi identity
t-deformation of inner product
 
Apr 4Schur Q-functions
Apr 6Projective representation theory
 
Apr 11Clifford algebras
Apr 13Guest lecture: Daniel Erman
 
Apr 18no class
Apr 20The basic spin representation
 
Apr 25Frobenius characteristic map for projective representations
Apr 27Schur Q-functions: wrapup
 
May 2no class
May 4Guest lecture: Jordan Ellenberg