Block of a group algebra

A block of a finite group algebra is an indecomposable two-sided ideal $RGe$ determined by a primitive central idempotent $e$. An $RG$-module $M$ lies in this block when $eM=M$.

A defect group of a block idempotent $e$ is a maximal p-subgroup $D$ for which the Brauer image ${\mathrm{Br}}_D(e)$ is nonzero. Equivalently, the vertices of the block algebra as an $R[G\times G]$-module are the diagonal subgroups $\Delta D$ for the defect groups $D$. All defect groups of a block are conjugate.

For a p-subgroup $D\le G$, let $D$ act on $kG$ by conjugation. The Brauer morphism is the algebra homomorphism

Let $D$ be a p-subgroup of $G$, put $N=N_G(D)$, and let $k$ have characteristic $p$. The Brauer morphism intertwines the two relative traces:Indeed, $D$ acts on $G/D$ by left multiplication. A coset $gD$ is fixed exactly when $g\in N$, and every other orbit has size divisible by $p$. After applying ${\mathrm{Br}}_D$, the summands belonging to one such orbit are equal, so every nonfixed orbit contributes zero in characteristic $p$; the fixed cosets give the trace from $D$ to $N$.

Let $B$ be a block of $kG$ with defect group $D$ and put $N=N_G(D)$. Its Brauer correspondent is the unique block $b$ of $kN$ with defect group $D$ selected by the nonzero Brauer image of the block idempotent of $B$.

Brauer's first main theorem gives a bijection between the blocks of $kG$ with defect group $D$ and the blocks of $k{N}_G(D)$ with defect group $D$. Corresponding blocks are related by their images under the Brauer morphism.

Over a splitting field of characteristic five, $A_5$ has a principal block of defect $C_5$ containing the ordinary characters of degrees $1,3,3,4$, and one defect-zero block containing the ordinary character of degree $5$. If $P$ is a Sylow 5-subgroup, then $N_{A_5}(P)\cong D_{10}$ and its unique 5-block is the Brauer correspondent of the principal block of $A_5$.

The decomposition matrix records the multiplicities of simple modular representations in reductions of ordinary representations. If its rows are indexed by ordinary irreducible characters and its columns by irreducible Brauer characters, its entry $d_{\chi \varphi }$ is the decomposition number of $\varphi$ in the reduction of $\chi$.

The Cartan matrix records composition-factor multiplicities in the projective indecomposable modules. For a split modular group algebra, it iswhere $D$ is the decomposition matrix.