REU Meeting - 2025-06-26

This following is a brief summary of our research meeting on 2025-06-26.

Meeting summary

This was our first meeting of the summer. We chatted about the general overview of the project, noting that 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.?

Tasks for next meeting

1. Summarize the 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.
2. 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
3. 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.)
4. 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

References

Dummit & Foote: Section 18.1 (for representation theory basics) and 7.2 (for an intro to group rings)