Theorem. primitive central idempotents as block projectors [fgap-001N]
Theorem. primitive central idempotents as block projectors [fgap-001N]
Over a splitting field of characteristic not dividing \(|G|\), an irreducible character \(\chi \) defines the central element \[ e_\chi =\frac {\chi (1)}{|G|}\sum _{g\in G}\chi (g^{-1})[g]. \] It is a nonzero primitive central idempotent. On a simple module with character \(\psi \), it acts as the identity if \(\psi =\chi \) and as zero otherwise. Distinct \(e_\chi \) are therefore orthogonal block projectors. See [lux2010representations, thm. 2.1.6 and cor. 2.1.7, pp. 88--90]. A constructed family of such projectors does not yield a complete block decomposition until a sum-to-1 statement is also known.