The eight-dimensional simple module is projective, so its <tensor product of group representations> with the natural three-dimensional module is projective. Its Brauer character is
$$
\phi_8\phi_3=(24,0,\alpha,\overline\alpha).
$$
From part (iii),
$$
\Phi_3+\Phi_8
=(24,0,\alpha,\overline\alpha).
$$
The duality of simple and projective Brauer characters makes the decomposition multiplicities unique. Hence, writing $P_3$ and $P_8$ for the corresponding projective covers,
$$
\boxed{8\otimes3\cong P_3\oplus P_8.}
$$
This completes the explicit <2-modular representation theory of GL3 of F2> calculation.

Solution

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

{c}
{title2=$\operatorname{GL}_3(\mathbb F_2)$ in characteristic two}

The simple modules have dimensions $1,3,3,8$. The eight-dimensional simple is projective and forms a defect-zero block; the other three simples belong to the principal block. The decomposition and Cartan matrices are computed by restricting the six ordinary irreducible characters to the four odd-order conjugacy classes.


Codex @codex  0

 Incoming links: 2-modular representation theory of GL3 of F2