Summer REU 2025 - Categorical representation theory

Project summary

We have three main aims:

1. Give categorical definitions for each object and property in classical representation theory.
2. Give categorical descriptions (and possibly proofs) for the various results (especially the named theorems) in representation theory.
3. Find a categorical interpretation/method of performing "calculations" in representation theory; e.g., computing character tables, inner products, etc.

Each of these aims is open-ended. Some things will be quickly and easily dealt with; e.g., reinterpreting representations of a group $G$ as functors from the "categorification" of $G$ to another category. Others might take some creativity; e.g., how do we define class functions, the trace, characters, etc.?

Meeting notes

Task list

Summarize Classical Representation Theory: Part I

• Summarize classical definitions of a representation of a group $G$, including:
  • matrix representations
  • $F$-linear representations, i.e., as actions on $F$-vector spaces (where $F$ is a field)
  • permutation representations
• Predict/conjecture analogous definitions for a representation of a group $G$ in other categories, such as:
  • topological representations
  • module representations
  • representations in your favorite category
• Review the concept of the group ring $R[G]$, where $G$ is a group and $R$ is a (commutative) ring. (One reference is Section 7.2 in Dummit & Foote)
• Prove there is a bijection (later to be upgraded to an equivalence of categories) between $F$-linear representations of a group $G$ and $F[G]$-modules
  • Also show that $F[G]$-submodules correspond to $G$-stable subspaces

Tasks will be added after each meeting.

The team

Aaron Boone

aboone03@calpoly.edu

Mark Muzquiz

mamuzqui@calpoly.edu

References

Dummit & Foote: Chapters 18 and 19