6
votes
1answer
263 views

What is the Grothendieck group of the category of $\mathbf{Z}_p[G]$-modules?

Let $G$ be a finite group. Let $\mathcal{O}$ be a suitably large finite extension of the $p$-adic integers, with residue field $\mathbf{F}_q$. The Grothendieck group of the category of ...
4
votes
2answers
543 views

Decomposing representations of finite groups

Let $G$ be a finite group, $p$ a prime number. We denote by $\mathbb{F}_p$ the field of cardinality $p$. Let $V$ be an infinite dimensional representation of $G$ over $\mathbb{F}_p$. Must there be ...
0
votes
1answer
134 views

Analogon to Brauer characters, if K not algebraically closed

Hello, is there a theory for characters of a finite group over a field $K$ with prime characteristic, if $K$ is not algebraically closed? For algebraically closed fields $K$ for example Brauer found ...
1
vote
1answer
222 views

Representations of semidirect product over $C_p$

Hi, I am wondering if anything is known about irreducible representations of a semidirect product over $C_p:=\mathbb{Z} / p \mathbb{Z}$ in general or at least in special cases. For example of $C_q ...
1
vote
1answer
254 views

Defect groups and subgroups

I have asked this question on Stack Exchange but had no response; it's been bugging me for a few days. I am struggling to see how to apply Mackey's theorem to prove a certain Lemma in Local ...
4
votes
2answers
679 views

two questions in modular representation theory

I have two questions: Let $G$ be a finite group. Because complex representations of $G$ are completely reducible to know all representations is same as knowing irreducible ones. In case of modules ...
6
votes
3answers
567 views

Exact sequences of permutational representations?

Let $R$ be a commutative ring, like the ring of integers $\mathbb Z$ or the ring of $p$-adic integers $\mathbb Z_p$. Let $G$ be a finite group; let us consider permutational representations of $G$ ...