REU Meeting - 2025-08-29

This following is a brief summary of our research meetings on 2025-08-29.

Meeting summary

Note: Aaron was out sick for this meeting.

We discussed the definition of a semisimple category and how it is mainly just helpful language, but probably not so helpful for proving anything. For example, Maschke's Theorem can be rephrased as saying that the category ${\mathbf{V}\mathbf{e}\mathbf{c}}_F^{\mathbf{G}}$ is semisimple when the characteristic of the field $F$ doesn't divide the order of the group $G$. The fact that there are any conditions, however, pretty much kills any hope that a category of the form $C^{\mathbf{G}}$ is semisimple whenever the category $C$ is semisimple. Alas.

We then talked a bit about the final share-out of the summer and how to organize our project in the write-up. We noted that our investigations into monoidal categories probably qualifies as an aside, at least for now, as we move onto our next big goal: Kan extensions and induced representations!

Tasks for next meeting

• Read Chapter X in Mac Lane, on Kan extensions. Focus mainly on Sections X.1, X.3, X.5 and X.7.
• Think about how to reframe induced representations and restrictions both as adjoints, and as an example of Kan extensions.
• Think about how the results on Kan extensions (especially when they exist and how to construct them) specifically in the case of induced reps. Can we give a categorical explanation for the construction of an induced representation?

References

Mac Lane, Categories for the Working Mathematician: Chapter X
Bump, Automorphic Forms and Representations: page 399 (induced reps)