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
$$
\boxed{
C=D^TD=
\begin{pmatrix}
2&1&1&0\\
1&3&2&0\\
1&2&3&0\\
0&0&0&1
\end{pmatrix}.}
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-138/5/c/ii/solution