Weight for a finite group

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 $$N_G(R)/R$$.