Math 445: Representation Theory

Syllabus

Click here for syllabus

Instructor

Name: Daniel Litt
Email: daniel.litt@utoronto.ca
Office Hours: 10am Wednesdays, Huron 1018

Teaching Assistant

Name: Austin Sun
Email: austin.sun@mail.utoronto.ca

Textbooks / Course Readings

Serre, Linear Representations of Finite Groups
Kirllov, Introduction to Lie Groups and Lie Algebras
Okounkov, Vershik, A new approach to representation theory of symmetric groups
Fulton and Harris, Representation Theory: A First Course

Notes, taken by Cole Franklin and Yun-chi Tang

See here: Notes

Notes from guest lectures

See here: Notes

Mark Breakdown

Component	Percentage
Assignments (best 4 of 5)	40%
Term Test	25%
Final Assessment	35%

Assignments

There will be 5 assignments. The assignment with the lowest grade will be dropped. Late submissions receive a mark of zero unless an extension is granted by the instructor. Tentative due dates are listed in the lecture schedule below.

Term Test

There will be one term test, which is 2 hours long, on February 27 in class.

Final Assessment

The final assessment will be held during the final assessment period in April 2025, scheduled by the registrar.

Schedule of Lectures

Week	Dates	Topics
Week 1	Jan 6 - Jan 10	Basics of representation theory of groups, semisimplicity, beginnings of character theory
Week 2	Jan 13 - Jan 17	Character theory, examples, induction and restriction, tensor products, HW 1 due
Week 3	Jan 20 - Jan 24	Group algebra, field of definition, Brauer’s theorem
Week 4	Jan 27 - Jan 31	Symmetric groups, HW 2 due
Week 5	Feb 3 - Feb 7	$\mathrm{SL}_2(\mathbf F_p)$
Week 6	Feb 10 - Feb 14	Lie algebras, $\mathfrak{sl}_2$, HW 3 due
Week 7	Feb 17 - Feb 21	Reading week (no classes)
Week 8	Feb 24 - Feb 28	$\mathfrak{sl}_2$ continued, and an application to combinatorics, Term Test (Feb 27)
Week 9	Mar 3 - Mar 7	More Lie algebras, ideals, root systems, HW 4 due
Week 10	Mar 10 - Mar 14	Classification of semisimple Lie algebras
Week 11	Mar 17 - Mar 21	Representations of semisimple Lie algebras
Week 12	Mar 24 - Mar 28	Pro-p groups, uniform pro-p groups, HW 5 due
Week 13	Mar 31 - Apr 4	Representation theory of $\mathrm{SL}_n(\mathbf Z)$, $n \geq 3$