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)(adsbygoogle = window.adsbygoogle || []).push({});

And now working with the case that you take: [itex] \Gamma_5 \otimes \Gamma_5[/itex] and want to find whether it's irreducible or reducible...

For that I need its character and using trace identities obtain:

[itex]X \equiv \chi_{\Gamma_5 \otimes \Gamma_5}= \chi_{\Gamma_5} \chi_{\Gamma_5}[/itex]

In this case so we have that:

[itex]X_2= X_4 =X_5 =0 [/itex]

and [itex]X_1 =X_3 = 4[/itex]

The equation proving wether it's reducible or not is:

[itex] \sum_i |C_i| X_i^2 = |G| = |D_4|=8 [/itex]

Else it would be greater than 8 (because of the existence of more than 1 Clebsch-Gordan coefficient)

This is true (since [itex]|C_1|=|C_3|=1[/itex] ) 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?

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# Characters and reducibility (D4 in particular)

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