4
$\begingroup$

In order to advance my research I'm supposed to understand this fact:

Let $A$ be an abelian group and $S$ a finite group acting on $A$. This defines the semi-direct product $A\rtimes S$:

Let $\chi$ be a character of $A$. Define $S_\chi = \{s\in S \mid s\chi = \chi\}$ (Recall: $s\chi = \chi \iff \forall a\in A, \chi(sas^{-1}) = \chi(a)$ ).

Note this defines a subgroup of $A \rtimes S> A \rtimes S_\chi$

We can extend each representation of $A$ to a representation of $A \rtimes S_\chi$ by: $\chi: A \rightarrow \mathbb{C}, \chi(as) := \chi(a)$.

Let $(\sigma,V) \in Irr (S_\chi)$. Note $\chi \otimes \sigma $ is a representation of $A \rtimes S_\chi$. Then:

  1. $Ind_{A \rtimes S_\chi}^{A \rtimes S} \chi \otimes \sigma$ is irreducible

  2. For any irreducible representation of $A \rtimes S$, there are $\chi \in Irr(A)$ and $\sigma \in Irr(S_{\chi})$ s.t. the representation is: $Ind_{A \rtimes S_\chi}^{A \rtimes S} \chi \otimes \sigma$

If I understand correctly the proof of this statement should be straight forward from Mackey theory, but I can't see the picture. My instructor calls this statement Mackey theory - which is different from the statement I see in other textbooks (seems like this is an application - correct me if I'm wrong).

Improve this question
$\endgroup$
4
  • $\begingroup$ I think you understand correctly, I find the “pedagogical” Chap. XII.1 of Fell-Doran (The Mackey Normal Subgroup Analysis for Finite Groups) a good place to see all statements and proofs in just about your desired generality (semidirect product simplifications in section 1.12). $\endgroup$
    – Francois Ziegler
    Commented Sep 7, 2019 at 14:46
  • 1
    $\begingroup$ Instead of Mackey, look for Clifford Theory. One possible reference is: G. Karpilovsky, Group Representations, Vol. 2, North-Holland Mathematics Studies, Volume 177 (North-Holland Publishing Company, Amsterdam, 1993). You can also look at the paper by Crisp, Onn and me, arxiv.org/abs/1607.04486, the beginning of Section 3 $\endgroup$
    – Ehud Meir
    Commented Sep 11, 2019 at 12:34
  • $\begingroup$ I have found the specific solution at: Linear· Representations of Finite Groups Serre, 8.2 Semidirect products by an abelian group $\endgroup$
    – Arnon Hod
    Commented Sep 12, 2019 at 10:07
  • $\begingroup$ A topological version of this (unitary representations of locally compact groups) is explained in Folland "A course in abstract harmonic analysis", page 201. $\endgroup$
    – Q-Zh
    Commented Nov 4 at 14:08

0

Reset to default

You must log in to answer this question.