Let $G=S_3$ be the symmetric group on three elements, whose character table is given as follows:
Let $V$ be the unique irreducible representation of dimension $2$
Question 1: Compute the character of the symmetric square representation $Sym^2(V)$
The character of a representation is the trace of a representation. How could one find the trace of $Sym^2V$?
Question 2: Decompose $Sym^2(V)$ into irreducible representations
How do I solve this problem using characters?
I know that the basis elements are:
$e_1^2=e_1 \otimes e_1$, $e_2^2=e_2 \otimes e_2$ and $e_1e_2=e_1 \otimes e_2 + e_2 \otimes e_1$
$\pi \otimes \pi$ acts trivially on $e_1e_2$, and $e_1^2$, $e_2^2$ give rise to $V$
So I believe we have: $Sym^2V \cong 1 \otimes V$
What I would like to know is: how could I have gotten this result by looking at characters
Thanks in advance for your help