Weight for a finite group

Definition

Definition with symbols

let $p$ be a prime and $G$ a finite group. A weight of $G$ is a pair $(R,\phi)$ such that:

• $R$ is a $p$-subgroup of $G$, and in fact, $R = O_p(N_G(R))$ (viz it is the $p$-Sylow core of its normalizer $N_G(R)$).
• $\phi$ is an irreducible character of $N_G(R)$, the restriction of $\phi$ to $R$ is trivial, and $\phi$ belongs to a $p$-block of $N_G(R)/R$ of defect zero.

Equivalence notion

Two weights are said to be equal if their is an inner automorphism of $G$ taking one to the other.

Weights for a block

For $B$ a $p$-block of $G$, the weight is said to eb a $B$-weight if $B=b^G$ where $b$ is the associated block on $N_G(R)/R$.