= Fong–Reynolds matrix decomposition
{c}
Let $N\triangleleft G$, let $b$ be a <block of a group algebra> of $kN$, and let $I$ be its inertia subgroup. A block of $kG$ covering $b$ is a full <matrix algebra> of degree $[G:I]$ over its corresponding block of $kI$. When $I=N$, the corner is just $kNb$. Let $e_B$ be the global block idempotent and put $c=e_Bb$. With coset representatives $g_i$, the elements $g_i c g_j^{-1}$ are matrix units in that global block because different conjugates of $b$ are orthogonal. When $I=N$, one has $c=b$.
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