# More than 1 character for a conjugacy class

1. Dec 2, 2014

### ChrisVer

Hi,
I am trying to find the characters for the $S_4$ group, in the conjugacy class $K= (..)(..)$ and the two 3 dimensional and one 2 dimensional representations of the group.

I use the relation obtained from the classes' algebra, which after some steps becomes::
$|K|^2 \chi^{\nu} (K) \chi^\nu (K) = d_\nu \Big[ 3 d_\nu + 2 |K| \chi^\nu (K) \Big]$
after inserting the numbers ($|K|=3, d=3$) I find for the two 3-dimensional the same equation, and give for those $X$'s:
$X(K)^2 - 2 X(K) -3 =0 \Rightarrow X=-1, X=3$

and for the 2-dimensional ($|K|=3, d=2$), $x$:
$3 x(K)^2 - 4 x(K) -4=0 \Rightarrow X=2, X= -\frac{2}{3}$

Obviously I'm getting the same result as the bibliography but I also get the $X=3$ and $x= - \frac{2}{3}$... How can I discard them?

2. Dec 7, 2014