REU Meeting - 2025-07-03

This following is a brief summary of our research meeting on 2025-07-03.

Meeting summary

Smooth sailing so far. Although everyone has hilariously chosen their own custom notation, there were no issues with categorifying a group $G$ and establishing isomorphisms between the category of $F$-linear representations of $G$, the category of $F[G]$-modules and the category ${\mathbf{\text{Vec}}}_F^{\mathbf{\text{G}}}$ of functors $\mathbf{\text{G}}\to {\mathbf{\text{Vec}}}_F$.

Aaron and Mark both cleanly (and simultaneously) outlined the natural transformations in ${\mathbf{\text{Vec}}}_F^{\mathbf{\text{G}}}$ and how they were equivalent to intertwiners of the corresponding $F$-linear representations (as well as module morphisms of the corresponding $F[G]$-modules).

We ended by sketching out the general battle plan for the next few meetings:

1. Summarize (with examples) the classical notions of irreducibility and indecomposability of representations, leading up to Maschke's Theorem and Schur's Lemma.
2. Port those classical notions into the category ${\mathbf{\text{Vec}}}_F^{\mathbf{\text{G}}}$.
3. Look for purely categorical methods to prove Maschke's Theorem and Schur's Lemma in ${\mathbf{\text{Vec}}}_F^{\mathbf{\text{G}}}$.

Tasks for next meeting

• classical definitions of
  • (ir)reducible representations
  • completely reducible representations
  • (in)decomposable representations
• Give explicit examples of each of the above , including possible combinations; e.g., a representation that is indecomposable and reducible.
• Give an example of a field extension $F\to E$ and a matrix representation over $F$ that is irreducible over $F$ but reducible over $E$ (see page 848 in Dummit & Foote).
  • Turn this into an example in the category of linear representations (as opposed to matrix representations).
• State Maschke's Theorem and summarize the proof
• Try the following exercises in Section 18.1 of Dummit & Foote:
  • 1, 2, 3, 5, 13-16, 20

References

Dummit & Foote: Section 18.1