Problem in representations of $D_4$ Problem
Let $G=D_4=\langle a,b\ |\ a^4=b^2=1,\ ab=ba^{-1}\rangle$ and $V=\mathbb{C}^2$ with bases $\{e_1,e_2\}$. We see $V$ as a $\mathbb{C}G-$module with action $ae_1=e_2,\ ae_2=-e_1$ and $be_1=e_1, \ be_2=-e_2$.

*

*Compute the character $\chi_V$ and show that the $\mathbb{C}G-$module $V$ is irreducible.

I found that $\chi_V(1)=2,\ \chi_V(a)=0,\ \chi_V(b)=0,\ \chi_V(a^2)=-2,\ \chi_V(ab)=0$ and that $[\chi_V,\chi_V]=1$ hence the $\mathbb{C}G-$module $V$ is irreducible.


*Examine if $V\otimes_{\mathbb{C}}V$ has a $\mathbb{C}G-$submodule isomorphic to $\mathbb{C}G$.

*For every $n\ge1$ we have the $\mathbb{C}G-$module $V_n=V^{\otimes n}$. Find all $n$ for which there exists $w\in V_n$ s.t. $w\not=0$ and $gw=w,\ \forall g\in G$.

I have to find $n$ s.t. $V_n^G=\{v\in V_n\ | \ gv=v\ \forall g\in G\}\not=\emptyset\iff dim_{\mathbb{C}}V_n^G\ge1$. I know that $dim_{\mathbb{C}}V_n^G=\dfrac{1}{|G|}\sum_{g\in G}\chi_{V_n}(g)=\dfrac{1}{|G|}\sum_{g\in G}[\chi_{V}(g)]^n=\begin{cases} 0, \ n\ \text{ is odd}\\ \ge1,\ n \ \text{is even} \end{cases}$. So it has to be $n\in2\mathbb{Z}$.

(a) Is my approach in $1,3$ correct?


(b) Any ideas for $2$?


(c) Is there a general method to see whether a module $A$ contains an isomorphic image of a module $B$ using characters? In particular how could I determine whether for a $\mathbb{C}G-$ module $V$,  $V\otimes_{\mathbb{C}} V$ contains a  $\mathbb{C}G-$ submodule isomorphic to $V$?

Thanks in advance!
 A: *

*Showing $\langle \chi_V, \chi_V\rangle = 1$ is fine.


*In general, $\langle \chi_V, \chi_{\mathbb{C} G}\rangle = \dim V$. This is just because $\chi_{\mathbb{C}G}(g) = |G| \delta_{g=1}$, so the scalar product simply picks off $\chi_V(1) = \dim V$. This example shows that indeed $\langle \chi_V, \chi_W\rangle$ need not tell you much of anything about whether or not $V$ contains $W$. It also shows that $V$ contains $\mathbb{C}G$ if and only if for all irreducibles $W$, $\langle \chi_V, \chi_W\rangle \geq \dim W$. This latter condition is of course easy to check if you know all $\chi_W$'s explicitly, and for such a small group you can work them out by hand.


*Computing $\frac{1}{|G|} \sum_{g \in G} \chi_V(g)^n$ is great.
In this particular case, it's easy to see $\langle \chi_V^2, \chi_V\rangle = 0$, so $V \otimes V$ doesn't contain the irreducible $V$, so it can't possibly contain $\mathbb{C}G$ and you don't have to check any other inequalities or hunt for other irreducible characters.
