What Changes in Representation Theory Over Non-Complex Fields?

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Discussion Overview

The discussion revolves around the implications of representation theory when applied to non-complex fields, particularly algebraically closed fields whose characteristic does not divide the order of the group. Participants explore the differences and similarities in character theory and representation when moving from complex fields to other well-behaved fields.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant notes the lack of resources on representation theory over non-complex fields and seeks guidance on what changes occur when the field is algebraically closed and the characteristic does not divide the group's order.
  • Another participant asserts that there are no significant changes in character degrees under these conditions, suggesting that the absence of literature reflects this simplicity.
  • A participant questions whether character degrees over the algebraically closed field \(\overline{\mathbb{F}_p}\) remain the same as those over \(\mathbb{C}\) when \(p\) does not divide \(|G|\).
  • One reply emphasizes that the characteristic of \(\mathbb{C}\) being zero is not crucial, as the relevant condition is the coprimality to \(|G|\). They reference a method by Brauer for transitioning from complex representations to those over fields of positive characteristic.
  • A separate question is raised regarding a specific result in representation theory about linear characters and Shoda pairs, with a request for assistance in proving a related statement.

Areas of Agreement / Disagreement

Participants express differing views on the implications of moving from complex fields to other algebraically closed fields. While some suggest minimal changes, others raise questions about specific aspects of representation theory that may not be straightforward.

Contextual Notes

Participants acknowledge the complexity of transferring results from the complex case to other fields, particularly concerning the role of characteristics and coprimality with group order. There is an unresolved question about the proof related to Shoda pairs and induced characters.

Who May Find This Useful

This discussion may be of interest to students and researchers in representation theory, particularly those exploring the application of the theory over non-complex fields and the nuances involved in such transitions.

Hello Kitty
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I'm done a basic course on representation theory and character theory of finite groups, mainly over a complex field. When the order of the group divides the characteristic of the field clearly things are very different.

What I'd like to learn about is what happens when the field is not complex but still quite well-behaved. In particular if we have an algebraically closed field whose characteristic doesn't divide the order of the group what changes?

The reason I ask is that there doesn't seem to be a very good treatment of this in any of the books I've seen. Can anyone offer any suggestions? I guess I could start from scratch and go though all the proofs in the complex case from the bottom up checking whether they still hold, but it would be nice to have a reference.

Are there any major pitfalls when trying to transfer the theory from the complex case?
 
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Hello Kitty said:
In particular if we have an algebraically closed field whose characteristic doesn't divide the order of the group what changes?
Nothing. That's why there are no treatises on the subject - the only added complication is it not being over C.
 
So the character degrees over \overline{\mathbb F_p} are the same as for \mathbb C provided p \not\vert \ |G|?
 
Look at your notes - where does it use that the characteristic of C is zero? It will only use that it is co-prime to |G|.

The method of going from C to char p was given by Brauer in the 50s. Reps over C are actually realizable over the algebraic integers, A. Pick a maximal ideal containing the prime ideal (p) in A, and reduce modulo this ideal. This yields the projective modules over the field of char p, which are all the modules if p is coprime to |G|.
 
I have a question in representation theory. There is a result that says that if I have a linear character of a subgroup H of a group G with kernel K, then the induced character is irreducible iff (H,K) is a Shoda pair.

The proof uses the fact that
If, chi(ghg-1)=chi(h) for all h in H ∩ g(-1)Hg, then
[H,g]∩H ⊂ K.

I am not able to prove this one...can sumbody help??
 

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