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I am trying to understand some group theory. In the notes I am following, I am told:

Recall the representations of $\mathcal{Z}_2$:

Trivial: $\rho_0(e) = 1$, $\rho_0(a)$ = 1

(i) $\rho_1(e) = 1$, $\rho_1(a)$ = -1

(ii) $\rho_2(e) = diag(1,1)$, $\rho_2(a) = diag(-1,-1)$

We see that $\rho_2$ is a combination of $\rho_1$ on $\left( \begin{array}{c} x\\ 0\\ \end{array} \right)$ and $\rho_1$ on $\left( \begin{array}{c} 0\\ y\\ \end{array} \right)$.

This last statement I don't understand. How is it a combination of these things?

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They are saying that $\rho_2 = \rho_1 \oplus \rho_1$. This means that the matrix block form of $\rho_2$ is $\rho_1$ in the first block and $\rho_1$ in the second block. The choice of notation "$\oplus$" makes more sense when you know about $FG$-modules (where $F$ is a field and $G$ a finite group). Feel free to ask more questions in the comment.

Hope that helps,

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  • $\begingroup$ The notes defined the left G-module before hand, but I don't understand what the operation $\oplus$ actually does. It seems to take a rep $\rho_i$ and represent it using one of the basis vectors, then using those to make a d-dimensional matrix (out of the column vectors?). Is that right? Thanks for your help. $\endgroup$
    – Akoben
    Aug 26, 2014 at 11:24

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