Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 138 / 2 / c / Solution

Apply part (b) with $P=P_i$ and $U=S_j$. A homomorphism $P_i\to S_j$ kills $J(P_i)$ and therefore factors through the head $P_i/J(P_i)\cong S_i$. By Schur lemma over the splitting field,Consequently the two stated bases satisfy the Duality of simple and projective Brauer characters:

The form is nondegenerate directly: if $\varphi \ne 0$, thenEquivalently, in coordinates indexed by the p-regular conjugacy classes it is a positive diagonal Hermitian form with weights $1/|C_G(x)|$.