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Find the number of inequivalent two-dimensional complex representations of the group $Z_4$

Any hints will be greatly appreciated. Thank you all

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  • $\begingroup$ Do you know what the irreducible representations of $\mathbb{Z}_4$ are? $\endgroup$
    – angryavian
    Dec 18, 2013 at 22:11

1 Answer 1

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Hint: Using Maschke's Theorem, any such two-dimensional representation of $\mathbb{Z}_4$ is the direct sum of irreducible representations of $\mathbb{Z}_4$. You need to find the irreducible representations of the group (which all turn out to be one-dimensional because the group is abelian). Thus, any two-dimensional representation is the sum of two of these irreducible representations. The question then becomes a combinatorial one: how many ways can you choose two irreducible representations?

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    $\begingroup$ Why does no one like abelian plays? Because the characters are all one-dimensional! (laughter ensues) $\endgroup$
    – Tim Ratigan
    Dec 18, 2013 at 22:29
  • $\begingroup$ Very nice pun, +1! $\endgroup$
    – Nicky Hekster
    Dec 18, 2013 at 22:42
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    $\begingroup$ @TimRatigan Even though all the characters are one-dimensional, they can sometimes be complex! $\endgroup$
    – angryavian
    Dec 18, 2013 at 23:00

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