Course description
A sequel to MAT631 Algebraic Combinatorics, though the basic objects from that course are recalled as they are needed. The goal is to understand the representations of the symmetric group. Along the way we see a good deal of beautiful combinatorics and a decent amount of algebra.
Grading
- Assignments 30% — biweekly
- Project 20% — summary of a contemporary research paper
- Presentation 20% — 45–60 minutes on the project topic
- End-semester exam 30% — 180 minutes, written
An A requires at least 80% overall, though 80% does not automatically guarantee one.
Lectures · 21
- 01 Mon 04 Aug Introduction to the course
- 02 Fri 08 Aug Matrix representations, \(G\)-modules
- 03 Mon 11 Aug Maschke's theorem
- 04 Mon 18 Aug Commutant algebra
- 05 Fri 22 Aug Group characters
- 06 Mon 25 Aug Induced and restricted representations
- 07 Fri 12 Sep Partitions
- 08 Mon 15 Sep Specht modules
- 09 Fri 19 Sep Submodule theorem
- 10 Mon 29 Sep A basis for \(S^\lambda\)
- 11 Fri 03 Oct Branching rule and Young's rule
- 12 Mon 06 Oct Applications of the R–S algorithm
- 13 Fri 10 Oct Viennot's shadows
- 14 Fri 17 Oct Hook-length formula
- 15 Fri 24 Oct Determinant form of the hook-length formula
- 16 Mon 27 Oct Hillman–Grassl correspondence
- 17 Fri 31 Oct A basis of the ring of symmetric functions
- 18 Mon 03 Nov Schur functions
- 19 Fri 07 Nov Jacobi–Trudi identities
- 20 Mon 17 Nov Connections to the representations of \(S_n\)
- 21 Mon 01 Dec End-semester examination
Submissions, late work and collaboration
Submissions
- Due at 11:59 pm on the due date.
- Typed in LaTeX, submitted as PDF or printed.
- Begin each solution on a new page.
- State your sources at the top of each problem, even if you worked independently.
Late policy
Late submissions are penalised 25% per day. No extensions, except for approved medical leave covering the day before and the day of the deadline.
Collaboration and sources
- Work on the problems independently first, then collaborate — meaningful collaboration is encouraged.
- You must write up your own solutions.
- At the top of each problem write "Collaborators and sources:" followed by everyone and everything you consulted, or "none". Failure to acknowledge costs 20% of that problem.
- Fine: looking up a standard theorem, formula or technique; using Wolfram Alpha, Mathematica or Python for a calculation.
- Not fine: looking up the problem itself online or in the literature, or using AI to solve the problem. Once you have solved it, seeking other solutions is welcome.
Intentional violations may be treated as academic misconduct.