A block induced from a subgroup

Fix a splitting $p$-modular system for finite $G$, with residue field $k$, and $H\le G$. For a primitive central idempotent $e$ of $kH$, write $b=kHe$ as the indecomposable double-group module of Block bimodule for the double group. A block $B$ of $kG$ is induced from $b$, denoted $b^G=B$, if it is the unique block for which $b\mid {\mathrm{Res}}_{H\times H}^{G\times G}B\mathrm{.}$ Here $X\mid Y$ means there are module maps $i:X\to Y$, $r:Y\to X$ with $ri={\mathrm{id}}_X$; equivalently $Y\cong X\oplus \ker r$.

Blocks are the actual ideals belonging to p-blocks from primitive central idempotents, not a choice of isomorphic copies. Distinct blocks $kGf,kG{f}^{\mathrm{\prime }}$ are nonisomorphic as bimodules: left multiplication by $f$ is the identity on the first and zero on the second, and every bimodule isomorphism would intertwine these operators. The finite multiplicities of indecomposable summands are well-defined by Finite-dimensional kG-modules decompose as finite direct sums of indecomposables uniquely up to order and isomorphism.

If no block, or more than one block, has the splitting property, $b^G$ is undefined. The definition never assigns a value in those cases. For $H=G$ the indecomposable block $b$ is its own unique such block, so $b^G=b$. No infinite family of choices is required by this definition. A central-character formula is a further theorem under additional hypotheses, not part of this definition.