# 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

### Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

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