Brauer correspondence

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 $B$.

The Brauer correspondent of a block of a group algebra with defect group of a block $D$ in $G$ is the corresponding block of $N_G(D)$ furnished by Brauer first main theorem. The more general induced block $b^G$ is defined by block induction when the subgroup contains the centralizer of a defect group of $b$.

If a block of a group algebra $b$ of $kH$ has defect group of a block $D$ and $D{C}_G(D)\le H\le G$, then its block induction $b^G$ is the principal block if and only if $b$ is the principal block. Equivalently, the Brauer morphism of the principal block idempotent at any p-subgroup $Q$ is the principal block idempotent of $k{C}_G(Q)$.

Let $b$ be a block of a group algebra of $kH$ with defect group of a block $D$, where $D{C}_G(D)\le H\le G$. Apply Brauer first main theorem inside $H$ to obtain its correspondent in $N_H(D)$. Composing that correspondent's central character of a block with the Brauer morphism from $Z(kG)$ selects a unique block of a group algebra $b^G$. It is the unique block of a group algebra whose restriction to $H\times H$ contains the block bimodule $b$ as a direct summand. Its defect group of a block can be larger than $D$ when $H$ does not contain $N_G(D)$.

The centralizer condition defining block induction refers to a defect group of a block of the actual source block. Containment of the centralizer of an unrelated p-subgroup does not suffice. For example, in characteristic $3$ take $G=A_5$, $H=A_4$ and $D=C_3$. Then $D{C}_G(D)\le H$, but the unique defect-zero block of $k{A}_4$ occurs on bimodule restriction of both three-dimensional defect-zero blocks of $k{A}_5$. The two ordinary degree-three characters have identical restriction to $A_4$, namely its degree-three character, and defect-zero reduction gives the same simple projective module. Thus the unique-block characterization fails if $D$ is not the defect group of the source block.

For a central idempotent $e$, a module $M=eM$, and $C_G(D)\le K\le N_G(D)$, put $b={\mathrm{Br}}_D(e)$. Then $M{\downarrow }_K=bM\oplus (1-b)M$, and every indecomposable summand of the second term has vertex not containing $D$. Indeed $e-b$ is a sum of $K$-orbit sums of group elements not centralizing $D$. Each is a relative trace from a centralizer not containing $D$. Applying these traces to the module and using the local endomorphism ring of an indecomposable summand proves relative projectivity for one such subgroup.

For an indecomposable $kK$-module $V$ with vertex $D$ and ${\mathrm{Br}}_D(e)V=V$, every indecomposable summand of $(1-e){\mathrm{Ind}}_K^{G}V$ has vertex conjugate into $D\cap {}^{g}D$ for some $g\notin N_G(D)$. The refinement keeps the intersection subgroups from Green correspondence, rather than asserting only that the remaining vertices are smaller than $D$.

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 for finite group representations, blocks of a group algebra with defect group of a block $D$ correspond to $N_G(D)$-orbits of blocks $b_D$ of $k(D{C}_G(D))$ with defect $D$ whose stabilizer quotient $N_G(D,b_D)/(D{C}_G(D))$ has order coprime to $p$. Equivalently one uses the defect-zero block of $k(D{C}_G(D)/D)$. This quotient is the inertial quotient of a block.

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 $A_5$.