Summer REU 2026 - Categorical representation theory

Project summary

Last summer we saw how most of the basic structures of representation theory can be re-contextualized in the setting of category theory. However, one of the biggest tools of classical representation theory is character theory and we haven't yet seen a categorical interpretation for characters. As such, our immediate aims are to:

1. Find a categorical interpretation of characters.
2. Find a categorical interpretation/method of performing "calculations" in representation theory; e.g., computing character tables, inner products, etc.
3. Look for purely categorical proofs of classical results involving characters.

Meeting notes

Task list

Tasks will be added after each meeting.

Classical character theory

• Read Section 18.3 in Dummit & Foote and check out the following exercises:
  • Exercise 1
  • Exercise 2
  • Exercise 5
  • Exercise 6
  • Exercise 7
  • Exercise 8
  • Exercise 11
  • [?] Exercise 12
  • [?] Exercise 13
• Read Section 18.2 in Dummit & Foote and check out the following exercises:
  • Exercise 1
  • Exercise 5
  • Exercise 8
  • Exercise 11

Categorical trace

• Learn about dualizable objects in monoidal categories. (One reference is here)
• Learn about the notion of a categorical trace. (A short reference is at Wikipedia. A longer reference is this paper by Ponto and Shulman.)
• Verify the following claims:
  • In the category of representations $\mathbf{\text{Rep}}(G)$, the monoidal unit $\mathbf{1}$ is the trivial representation $k$.
  • The endomorphisms of the trivial representation are just scalars: $\mathrm{End}(k)\cong k$.
  • Taking the categorical trace of an endomorphism of a representation yields a scalar, which precisely recovers the value of the character.

Categorical characters: Part I

• Skim through this nLab page about actions on monoidal categories.
• Try to make sense of this claim from Gemini:
• Check out this paper by Kapranov and Ganter
• Unwind the definition of categorical characteristic

Tannaka duality

• Learn about Tannaka duality by skimming through this nLab page
• Learn about categorical ends
• Explain how Tannaka's reconstruction theorem is "just enriched Yoneda in slight disguise"

Categorical characters: Part II

• With our new knowledge, try to express classical characters from a purely categorical perspective

Wrap-up: Final thoughts

• Try to explain in your own words how the classical isomorphism of $G$ with its character group is replaced/generalized by Tannaka duality
• "Prove" that result by adapting our "enriched Yoneda" proof of Tannaka
• Continue writing up what we've learned/done
• Think about the poster

The team

Mark Muzquiz

mamuzqui@calpoly.edu

References

Dummit & Foote, Abstract Algebra (Chapters 18 and 19)
Mac Lane, Categories for the Working Mathematician