REU Meeting - 2025-07-07

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

Meeting summary

Aaron was out of commission for this meeting, so we slowed down a bit. We chatted about the notions of (ir)reducible and (in)decomposable, and noted how the two concepts coincide in characteristic zero by Maschke's Theorem. Speaking of which, we spent some time outlining the proof of Maschke's Theorem, including a diagram indicating how the argument is really about moving back and forth between two categories, namely the category ${\mathbf{\text{Vec}}}_F$ of $F$-vector spaces and the category $F[G]\mathbf{\text{-Mod}}$ of $F[G]$-modules.

We then spent a few minutes briefly going over the solutions for Exercises 1, 2, 3, 5 and 13 in Section 18.1 of Dummit & Foote, and ended with a sketch of how we'll tackle the notion of subobjects in general categories (soon). First, though, we'll focus on the last batch of exercises from Section 18.1 in Dummit & Foote.

Tasks for next meeting

Finish off the following tasks from last time:

• Give explicit examples of representations with various reducibility/decomposability properties, 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).
• Try the following exercises in Section 18.1 of Dummit & Foote:
  • 14-16, 20

References

Dummit & Foote: Section 18.1