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I have got some problems such as:"Decompose the real representation $A$ of the group $G$ into a direct sum of irreps". Following problem is one of them.

Let $\Phi$ be the real representation of the cyclic group $C_4,$ such as $ \Phi(c)=\begin{bmatrix}0 & 0 &-1\\0 & 1 & 0 \\1 & 0 & 0\end{bmatrix}.$

Decompose $ \Phi$ into a direct sum of irreps.

I know real irrreps of the cyclic group $C_4. They are:

  • trivial representation $T$
  • representation $S$, such as $S(c)=S(c^3)=-1,$ $S(e)=S(c^2)$
  • two-dimensional representation $A,$ such as $A(c)=\begin{bmatrix}0 &-1\\ 1 & 0 \end{bmatrix}.$ I am not sure how to continue. Thanks a lot in advance for any help!
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  • $\begingroup$ Use the character formulas.... $\endgroup$ – Igor Rivin Jan 5 '14 at 23:38
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Swapping the second and third rows, and second and third columns changes your matrix to $\pmatrix{ 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0& 0 & 1} $. Can you now see how to decompose your representation?

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  • $\begingroup$ Maybe $\pmatrix{ 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0& 0 & 1} =T \oplus A.$ Why Can I swap rows and columns? $\endgroup$ – nadia-liza Jan 6 '14 at 0:02
  • $\begingroup$ Right. Because you can achieve it by conjugating by the permutation matrix corresponding to the permutation $(2,3)$. $\endgroup$ – Konstantin Ardakov Jan 6 '14 at 0:35

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