Math 122: Modules and Group Representation Theory

This class will meet on-line Tuesdays and Thursdays at 1:30 PDT, starting Tuesday April 7. Use Canvas to obtain a Zoom link to the lecture.

• Instructor: Daniel Bump (bump at math dot stanford dot edu)
• Course Assistant: Matt Larson
• My office hours will be Wednesdays and Fridays 12:30-2:30.
• Matt's office hours will be Mondays 6-8 PM (starting April 13) and 8:20-9:20 AM on Wednesdays.
• Office hours, like lectures, are scheduled Zoom meetings. Sign in through Canvas, and use the Stanford SSO to log into Zoom.

Syllabus

ExploreCourses syllabus: Modules over a principal ideal domain. Tensor products over fields. Group representations and group rings. Maschke's theorem and character theory. Character tables, construction of representations.

In Dummit and Foote, we will cover Sections 10.1-4 and Chapters 12,18,19. We will also review semidirect products from Section 5.5. I expect to emphasize induced representations more than Dummit and Foote do.

Announcements

During the week of April 13, Brian Conrad is doing the lectures. This is because of minor illness on my part, but has a benefit that the change in emphasis from the treatment of tensor products in Dummit and Foote will be an improvement. I expect to resume teaching next Tuesday April 21.

Homeworks

• Homework 1: (Due Wednesday, April 15): Section 10.1 # 7,8,9,10,15; Section 10.2 # 10; Section 10.3 # 2,7,15. In two of the problems in Section 10.3, there are references to earlier problems that were not assigned. Treat these references as hints. You should be able to include enough details in your solutions that they are self-contained. Text of the problems.
• Homework 2 (Due Wednesday April 22.) Thanks to Brian Conrad for these problems.
• Homework 3 (Due Wednesday April 29.) In case you want to see the tex file, here it is: homework3.tex.
• Homework 4, Due Wednesday May 6. Dummit and Foote Section 18.1 # 13,14,15,16,18; 18.3 # 5,6,11.
• Homework 5, Due Wednesday May 13. Dummit and Foote Section 18.3 # 9,12,23,27, Section 19.1 # 3,5,8
• Homework 19.1 #4 was assigned this week but HW 5 was too long so I am going to postpone it. I will also talk about this in a lecture.
• Homework 6, Due Friday, May 22. Dummit and Foote Section 5.5 # 8. Section 18.2 # 16. Section 19.1 # 2,4,7, Section 19.3 #2 The first problem depends on semidirect product theory, so you may need to read a bit of Chapter 5, Section 5.
• Project on Induced Representations due Monday June 1 (extended from April 29).
Here is the latex file for the project. Corrected May 26. The first displayed formula on the second page should read: $f(h g) = \pi(h)(f(g))$. (A parenthesis was in the wrong place.) If this is not what you see refresh your browser cache.

Hints for the project:

Hint for Problem 2: find the slide called Definition of $\chi^G$ in Lecture 12. The $d_i$ in the project hint are the $m_i$ in the slide.

The hardest part may be Problem 3(d). First prove that any $f\in I(V)$ we have a formula:

$f=\sum_{i=1}^h\Pi(x_i^{-1})\hat v_i$

where $x_i$ are a set of right coset representatives for $G=\bigcup Hx_i$ and $v_i=f(x_i)$.

Then show that $\Theta$ must satisfy:

$\Theta(f)=\sum_i x_i^{-1}\theta(f(x_i))$

Lecture Notes

• Lecture 1: Module Basics (April 7). Here are the remarks made during the lecture.
• Lecture 2: Free Modules (April 9). This is the pdf of our Zoom meeting.
• Maps with Tensor Products (April 16). Notes by Brian Conrad. Read these instead of Section 10.4 in Dummit and Foote.
• To view the lecture by Brian Conrad on Thursday, April 16, go to the Canvas page for this class and look under Files.
• Lecture 5: More on Tensors (April 21).
• Lecture 6: Maschke's Theorem (April 23).
• Lecture 7: Characters (I) (April 28).
• Lecture 8: Schur Orthogonality (April 30). In class notes for this lecture.
• Lecture 9: Permutation Representations (May 5). In class notes for this lecture.
• Lecture 10: Wedderburn's theorem (I) (May 7).
• Lecture 11: Wedderburn's theorem (II) (May 12). In class notes for this lecture.
• Lecture 12: Induced Characters (May 14). In class notes for this lecture, completing the example of the dihedral representation that was unfinished in the prepared text.
• Lecture 13: Permutation Characters (II) (May 19).
• Lecture 14: Frobenius Groups (I) (May 21). In class notes for this lecture.
• Lecture 15: Frobenius Groups (II) (May 26).
• Lecture 16: Frobenius Groups (III) (May 26).
• Lecture 17: Modules over PID's (June 2).
• Lecture 18: Character degrees and canonical forms (June 4).
• Lecture 19: Burnside's $p^aq^b$ theorem (June 9).

Notes from a previous course

• Group representation theory