The rows of the <Cartan matrix of a group algebra> express the <projective characters> in the simple Brauer-character basis. Thus
$$
\begin{aligned}
\Phi_1&=2\phi_1+\phi_3+\phi_{\overline3},\\
\Phi_3&=\phi_1+3\phi_3+2\phi_{\overline3},\\
\Phi_{\overline3}&=\phi_1+2\phi_3+3\phi_{\overline3},\\
\Phi_8&=\phi_8.
\end{aligned}
$$
Evaluating gives
$$
\boxed{
\begin{array}{c|rrrr}
&1&3&7A&7B\\\hline
\Phi_1&8&2&1&1\\
\Phi_3&16&1&\alpha-1&\overline\alpha-1\\
\Phi_{\overline3}&16&1&\overline\alpha-1&\alpha-1\\
\Phi_8&8&-1&1&1
\end{array}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 138 / 5 / c / iii / Solution

On the 2-regular classes the six ordinary characters decompose as
$$
\chi_1=\phi_1,
\quad \chi_{3A}=\phi_3,
\quad \chi_{3B}=\phi_{\overline3},
\quad \chi_6=\phi_3+\phi_{\overline3},
\quad \chi_7=\phi_1+\phi_3+\phi_{\overline3},
\quad \chi_8=\phi_8.
$$
Therefore the <decomposition matrix (modular representation theory)>, with columns ordered $1,3,\overline3,8$, is
$$
\boxed{
D=\begin{pmatrix}
1&0&0&0\\
0&1&0&0\\
0&0&1&0\\
0&1&1&0\\
1&1&1&0\\
0&0&0&1
\end{pmatrix}.}
$$
The <Cartan matrix of a group algebra> is
$$
\boxed{
C=D^TD=
\begin{pmatrix}
2&1&1&0\\
1&3&2&0\\
1&2&3&0\\
0&0&0&1
\end{pmatrix}.}
$$

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 138 / 5 / c / ii / Solution

{title2=$C=D^TD$}

The Cartan matrix records composition-factor multiplicities in the projective indecomposable modules. For a split modular group algebra, it is
$$
C=D^TD,
$$
where $D$ is the <decomposition matrix (modular representation theory)>.

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