REU Meeting - 2026-07-02

This following is a brief summary of our research meeting on 2026-07-02.

Meeting summary

We went through solutions to those four assigned exercises from Section 18.2, skimming over a few details but hitting the main points. We still haven't finished off the last two exercises from Section 18.3, but we'll get to those eventually.

Tasks for next meeting

It turns out there's a categorical notion of trace (and perhaps even of characters!). For next time, let's try the following:

• 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.

After we get our minds wrapped around this, we can move onto a categorical notion of characters!

References

(embedded in the above tasks)