Math 201A - Representation Stability (Winter 2019)

Lecture times and location: MWF 2:00PM - 2:50PM, AP&M 5402
Instructor: Steven Sam email
Instructor office hours: AP&M 7220, TBA

This is an introduction to representation stability. This is a relatively new area which combines combinatorics, representation theory, algebraic geometry, topology, group theory, and other areas. Typical examples in the subject involve sequences of representations of groups (symmetric groups, general linear groups, wreath product groups, etc.) which exhibit some kind of stable pattern. The goal is to study some examples and develop some of the theory. It is intended for students who have had a first-year graduate course in abstract algebra. While the examples come from more advanced topics, I will try my best to minimize the prerequisites to understand them. There is no textbook for the course. Instead, I will maintain typed lecture notes, which can be found below. Some references you may find useful:

Representation theory and homological stability - paper by Church and Farb which first attempts to formalize representation stability. Contains many examples.
Benson Farb's ICM lecture on representation stability - overview of subject and has some examples
Lecture notes from symmetric functions, Spring 2017 - useful for more detailed representation theory calculations
Noetherian properties in representation theory - conference talk I gave illustrating the use of noetherianity in representation stability
Introduction to twisted commutative algebras - Twisted commutative algebras are some structures we will discuss. Also contains summary of representation theory of symmetric and general linear groups.

Notes for course

Notes from a similar course (semester-long) I taught at UW-Madison

Schedule


Jan 7	Introduction (Section 1)
Jan 9	FI-modules (definitions) Configuration spaces (Section 5)
Jan 11	Configuration spaces (Section 5) Spectral sequences (Appendix A.4)
Jan 14	Representations of categories (Section 3.4) Noetherian posets (Section 3.5)
Jan 16	Representations of categories (Section 3.4)
Jan 18	Finished proof of local noetherian property of FI-modules

Jan 21	No class
Jan 23	Cohomology of unordered configuration spaces (Section 5)
Jan 25	Complex of injective words (Section 4.1)

Jan 28	Nakaoka's theorem (Section 4.2)
Jan 30	Twisted homological stability for FI-modules (Section 4.3)
Feb 1	Modules over the divided power algebra (Section 7.5)

Feb 4	Twisted commutative algebras (Section 3.1)
Feb 6	Cohomology of FI-modules (Section 7.5)
Feb 8	Survey of other combinatorial categories

Feb 11	Guest lecture: Brendon Rhoades
Feb 13	Guest lecture: Brendon Rhoades
Feb 15	Guest lecture: Brendon Rhoades

Feb 18	No class
Feb 20	Serre quotient categories (Section 7.2)
Feb 22	Generic FI-modules (Section 7.3)

Feb 25	Stable Kronecker coefficients (Section 7.1) Border rank of tensors (Section 6.2)
Feb 27	Draisma-Kuttler theorem (Section 6.4)
Mar 1	Draisma-Kuttler theorem Polynomial functors

Mar 4	no lecture
Mar 6	Bounded polynomial functors
Mar 8	Δ-modules (Section 8)

Mar 11	Minimal free resolutions
Mar 13	Syzygies of Segre embeddings
Mar 15	Koike-Terada modification rules