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

A block idempotent is a primitive central idempotent $e\in R$. The module $M$ lies in the corresponding block of a group algebra whenequivalently when every other block idempotent annihilates $M$.

If $V$ lies in $e$, then the centrality of $e$ makes both its submodule $U$ and quotient $W$ lie in $e$. Conversely, suppose $eU=U$ and $eW=W$. Then $(1-e)V$ maps to zero in $W$, so $(1-e)V\subseteq U$. But $(1-e)U=0$, and applying the idempotent $1-e$ once more givesThus $eV=V$, proving

Regard the block algebra $RGe$ as an $R[G\times G]$-module through left and right multiplication,A defect group of a block is a p-subgroup $D\le G$ for which $\Delta D=\{(d,d):d\in D\}$ is a vertex of an indecomposable summand determining the block; equivalently, $D$ is maximal withunder the Brauer morphism. The uniqueness of vertices up to conjugacy in $G\times G$, together with the diagonal form of these vertices, shows that any two such $D$ are conjugate in $G$. Hence

Only the classes $1,3,7A,7B$ are 2-regular. The trivial module and the natural three-dimensional module are simple; the dual natural module gives the conjugate three-dimensional character. Restricting the ordinary characters to the odd-order classes and using $\alpha +\overline{\alpha }=-1$ produces the fourth simple character of degree eight. The Brauer character table isThe ordinary degree-eight character restricts exactly to ${\varphi }_8$. Since its degree contains the full 2-part $8$ of $|G|$, this simple module is projective and its singleton block has defect zero. Thus

On the 2-regular classes the six ordinary characters decompose asTherefore the decomposition matrix, with columns ordered $1,3,\overline{3},8$, isThe Cartan matrix of a group algebra is

The rows of the Cartan matrix of a group algebra express the projective characters in the simple Brauer-character basis. ThusEvaluating gives

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 isFrom part (iii),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,This completes the explicit 2-modular representation theory of GL3 of F2 calculation.