Characters and reducibility (D4 in particular)

[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?

 

[jvicens]

Based on the information provided, it seems that the representation \Gamma_5 \otimes \Gamma_5 is irreducible and does not coincide with any of the 5 known irreps of D4. This means that it is a new and unique irrep for D4. However, to fully determine its properties and equivalence to other irreps, more information and analysis would be needed. It is possible that this irrep may have similarities or connections to other known irreps, but further investigation would be necessary to determine this.