# Characters and reducibility (D4 in particular)

1. Nov 28, 2014

### ChrisVer

I am working with D4 group . I have found its character table (check Table 4 here: http://eprints.utm.my/63/1/Irreducible_Representations_of_Groups_of_Order_8.pdf)

And now working with the case that you take: $\Gamma_5 \otimes \Gamma_5$ and want to find whether it's irreducible or reducible...
For that I need its character and using trace identities obtain:
$X \equiv \chi_{\Gamma_5 \otimes \Gamma_5}= \chi_{\Gamma_5} \chi_{\Gamma_5}$
In this case so we have that:
$X_2= X_4 =X_5 =0$
and $X_1 =X_3 = 4$

The equation proving wether it's reducible or not is:
$\sum_i |C_i| X_i^2 = |G| = |D_4|=8$
Else it would be greater than 8 (because of the existence of more than 1 Clebsch-Gordan coefficient)
This is true (since $|C_1|=|C_3|=1$ ) so the representation is irreducible...
However its characters are =4 and they don't coincide with any other character of the 5-known-irreps of D4... to which irrep is it equivalent?

2. Dec 3, 2014