= 2-modular blocks of S3
{c}
Over a <splitting field for finite group representations> of characteristic $2$, put $a=(123)$. The <block of a group algebra> idempotents of $kS_3$ are $b_0=1+a+a^2$ and $b_1=a+a^2$. Their block algebras are $b_0kS_3\cong kC_2$ and $b_1kS_3\cong M_2(k)$, with a <defect group of a block> given by $C_2$ and $1$, respectively. The first is a <local ring>; for the second, the representation $a\mapsto\operatorname{diag}(\omega,\omega^2)$, $(12)\mapsto\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)$ generates the full matrix algebra. The <Cartan matrix of a group algebra> is $\operatorname{diag}(2,1)$, and the ordinary trivial, sign, and two-dimensional characters give <decomposition matrix (modular representation theory)> $\left(\begin{smallmatrix}1&0\\1&0\\0&1\end{smallmatrix}\right)$.
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