Brauer correspondence for a defect-D8 block of S7 in characteristic 2

Write $s=(12)$ and $r=(1324)$. Then $r^4=s^2=1$, $srs=r^{-1}$ and $s\mathrm{\notin }⟨r⟩$, giving exactly the eight elements $r^j,s{r}^j$. A normalizer preserves the common fixed set $\{5,6,7\}$, so it is $N_{S_4}(D)\times S_3$. The first factor has order $8$ or $24$ by divisibility. It is not $S_4$: conjugating $(12)$ to $(13)$ leaves $D$, whose only transpositions are $(12)$ and $(34)$. Therefore it is $D$.

In the last $S_3$ write $a^3=t^2=1$, $tat=a^{-1}$. The element $e_0=1+a+a^2$ is a central idempotent in characteristic $2$. Its ideal has basis $e_0,t{e}_0$ and is $k{C}_2$, since $(t{e}_0{)}^2=e_0$. The complementary idempotent is $e_1=a+a^2$. To identify its four-dimensional ideal, represent $a$ by $A=\left(\begin{array}{cc}0& 1\\ 1& 1\end{array}\right)$ and $t$ by $T=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$. Direct multiplication gives $A^3=T^2=I$, $TAT=A^2$ and $I+A+A^2=0$. The image contains $E_{22}=A+T$, $E_{11}=I+E_{22}$, $E_{21}=AT+I$, and $E_{12}=T+E_{21}$. Thus it is all $M_2(k)$; dimension gives $k{S}_3{e}_1\cong M_2(k)$.

Hence $kN\cong k[D\times C_2]\times M_2(kD)$. F4 makes the first factor local and gives no nontrivial idempotents in $kD$. A central matrix commuting with every matrix unit is a scalar matrix over $Z(kD)$; thus the second factor has no nontrivial central idempotents either. These are exactly the two blocks. Augmentation is $1$ on $e_0$ and $0$ on $e_1$, so F3 places the trivial module in the first, the principal block. Its defect is $D\times ⟨t⟩$ by F2.

Regard $e_1=a+a^2$ as an element of $kN$. Both support elements centralize $D$, so ${\mathrm{Br}}_D(e_1)=e_1\mathrm{\ne }0$. If a $2$-subgroup $R$ of $N$ properly contains $D$, its projection to $S_3$ is a subgroup of order $2$. Its involution centralizes neither $a$ nor $a^2$. Thus neither support element centralizes $R$ and ${\mathrm{Br}}_R(e_1)=0$. F5 and F6 prove that $D$ is a defect group of the second block.

F1 now gives exactly one global block having $D$ as a defect group, paired with this nonprincipal local block. It is not the global principal block: the $2$-part of $7!=5040$ is $16$, whereas $\mathrm{\mid }D\mathrm{\mid }=8$, and F2 gives Sylow defect for the principal block. The larger local principal defect has order $16$ as well. All idempotents and matrix units were displayed; there is no zero block, empty correspondence, endpoint parameter or choice operation in this calculation. The correspondence is bijective in both directions by F1.