Let $V$ be a vector space over $F$.
Let $\varphi:G\to GL(V)$ be a representation.
If $G$ is infinite or $\text{char }F$ divide $|G|$ or $\dim{V}=\infty$,
then an irreducible representation of $G$ is still indecomposable?
Could anyone help me complete the table by give an example in each case?
(Or point out the mistakes in the chart.)
I am not familiar with the representation of infinite groups.
The examples are given as easy as possible.
Please avoid using module (if possible).
Thanks.
The representation $\varphi:\Bbb{Z}\to GL_2(\Bbb{C})$, $\varphi(n)=\left(\begin{smallmatrix} 1 & n \\0 & 1 \end{smallmatrix}\right)$ is not completely reducible see here.
Best Answer
Here are some examples. The page number refer to Dummit and Foote's Abstract Algebra 3/e. The gray region means that no module can satisfy the condition.
Focus on
the suffix of the term "decomposable" and
the first word of the term "completely reducible".
Decomposable just means that it can be able to be decomposed.
Completely reducible means that it can not only be reduced but also this reduced process can be done continuously until it is reduced completely.
This note may help someone to clarify the difference between decomposable and completely reducible.
Note that indecomposable doesn't imply irreducible.
Note that decomposable doesn't imply completely reducible.