I am reading the following textbook:
Representations and Characters of Groups, Gordon James & Martin Liebeck p.50
In the example 5.5:
Let $G = C_3 = \langle a:a^3=1\rangle$ and let $V$ be the $3$-dimensional $FG$-module with basis $v_1,v_2,v_3$ such that $$v_1a=v_2, v_2a=v_3, v_3a=v_1$$$V$ is a reducible $FG$-module, and has an $FG$-submodule $W$ = span$(v_1+v_2+v_3)$.
Let $\mathcal{B} = \{v_1+v_2+v_3,v_1,v_2\}$ of $V$. Then
which is the matrix representation of $1,a,a^2$ based on the basis $\mathcal{B}$.
My question is how to find these matrices. I know the top-left block which is always $1$ since this represents $v_1+v_2+v_3$. Top-right is $0$ since $V$ is reducible. But how about other blocks?
We have for instance formally, but i hope that it is clear how to extend matrix computations over a ring / field to the analogous when we have a mixed / external operation of a (group) ring on some module... $$ \begin{aligned} \begin{bmatrix} v_1+v_2+v_3\\v_1\\v_2 \end{bmatrix} \cdot 1 &= \begin{bmatrix} (v_1+v_2+v_3)\cdot 1\\v_1\cdot 1\\v_2\cdot 1 \end{bmatrix} = \begin{bmatrix} v_1+v_2+v_3\\v_3\\v_1 \end{bmatrix} \\ &= \color{blue}{ \begin{bmatrix} 1 &&\\&1&\\&&1 \end{bmatrix}} \begin{bmatrix} v_1+v_2+v_3\\v_1\\v_2 \end{bmatrix}\ , \\[3mm] % ========== \begin{bmatrix} v_1+v_2+v_3\\v_1\\v_2 \end{bmatrix} \cdot a &= \begin{bmatrix} (v_1+v_2+v_3)\cdot a\\v_1\cdot a\\v_2\cdot a \end{bmatrix} =\begin{bmatrix} v_1+v_2+v_3\\v_2\\v_3 \end{bmatrix} \\ &= \begin{bmatrix} v_1+v_2+v_3\\v_2\\(v_1+v_2+v_3)-v_1-v_2 \end{bmatrix} \\ &= \color{blue}{ \begin{bmatrix} 1 &&\\&&1\\1&-1&-1 \end{bmatrix}} \begin{bmatrix} v_1+v_2+v_3\\v_1\\v_2 \end{bmatrix}\ , \\[3mm] % ========== \begin{bmatrix} v_1+v_2+v_3\\v_1\\v_2 \end{bmatrix} \cdot a^2 &= \begin{bmatrix} (v_1+v_2+v_3)\cdot a^2\\v_1\cdot a^2\\v_2\cdot a^2 \end{bmatrix} =\begin{bmatrix} v_1+v_2+v_3\\v_3\\v_1 \end{bmatrix} \\ &= \begin{bmatrix} v_1+v_2+v_3\\(v_1+v_2+v_3)-v_1-v_2\\v_1 \end{bmatrix} \\ &= \color{blue}{ \begin{bmatrix} 1 &&\\1&-1&-1\\&1& \end{bmatrix}} \begin{bmatrix} v_1+v_2+v_3\\v_1\\v_2 \end{bmatrix}\ . \end{aligned} $$