Syllabus

The topics covered in each class will be given in the table below, along with the corresponding sections of the notes.

Date Notes Topics
Jan 10 1.1–1.3 The symmetric group, partitions and compositions, graded rings
Jan 12 1.3, 1.4 Graded rings (cont.), formal power series
Jan 17 1.4, 2.1, 2.2 Formal power series (cont.), symmetric polynomials, symmetric functions
Jan 19 2.2, 2.3, 2.4 Symmetric functions (cont.), tableaux, elementary symmetric functions
Jan 24 2.4, 2.5 Elementary symmetric functions (cont.), homogeneous symmetric functions
Jan 26 2.6 Power sums
Jan 31 3.1, 3.2 Alternating functions, Schur functions
Feb 2 3.2, 3.3 Jacobi–Trudi identities, the Hall inner product
Feb 7 3.3, 3.4 The Hall inner product (cont.), skew Schur functions
Feb 9 3.4 Tableaux description of skew Schur functions
Feb 14 3.5 Transition matrices
Feb 16 3.6 The Littlewood–Richardson rule
Feb 21–25 Reading week
Feb 28 4.1, 4.2 Tensor products, adjoint operators
Mar 2 4.2, 4.3 Adjoint operators (cont.), Hopf algebras
Mar 7 4.3, 4.4 Hopf algebras (cont.), Hopf algebra structure on the ring of symmetric functions
Mar 9 4.4, 3.7 Hopf algebra structure on the ring of symmetric functions (cont.), the Murnaghan–Nakayama rule
Mar 14 5.1, 5.2 The Heisenberg algebra, bosonic Fock space
Mar 16 5.3 Fermionic Fock space
Mar 21 5.4 Bosonization
Mar 23 5.5 Fermionization
Mar 28 6.1, 6.2 Characters of finite groups, induction and restriction
Mar 30 6.3 Characters of the symmetric group
Apr 4 6.4 Specht modules
Apr 6 6.5 Representations of the general linear group