Exploring Representation Theory in Non-Complex Fields

In summary, if you 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.
  • #1
Hello Kitty
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0
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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  • #2
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.
 
  • #3
So the character degrees over [tex] \overline{\mathbb F_p} [/tex] are the same as for [tex] \mathbb C [/tex] provided [tex]p \not\vert \ |G| [/tex]?
 
  • #4
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|.
 
  • #5
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??
 

1. What is representation theory?

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing them as linear transformations of vector spaces. It is used to understand the structure and properties of algebraic objects such as groups, rings, and algebras.

2. What is the importance of representation theory?

Representation theory has a wide range of applications in various areas of mathematics, physics, and engineering. It helps in understanding symmetry and in solving problems related to algebraic structures. It is also used in coding theory, signal processing, and quantum mechanics.

3. What are the different types of representations?

The two main types of representations in representation theory are finite-dimensional representations and infinite-dimensional representations. Finite-dimensional representations are those that can be written as matrices, while infinite-dimensional representations involve infinite-dimensional vector spaces.

4. Can representation theory be applied to real-world problems?

Yes, representation theory has many practical applications in fields such as physics, chemistry, and computer science. For example, it is used in quantum mechanics to study the symmetries of particles and their interactions. In chemistry, it is used to understand the structure and properties of molecules.

5. How is representation theory related to other branches of mathematics?

Representation theory is closely related to other branches of mathematics such as group theory, linear algebra, and abstract algebra. It also has connections to topology, differential equations, and combinatorics. Many important concepts in representation theory have been developed by drawing ideas from these other areas of mathematics.

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