Projective indecomposable modules, Scott modules and the Frobenius–Schur indicator

Let Φ be a principal indecomposable character of a finite group G in characteristic 2. The Frobenius–Schur indicator ν(Φ) of Φ is shown to equal the rank of a bilinear form defined on the span of the involutions in G. Moreover, if the principal indecomposable module corresponding to Φ affords a quadratic geometry, then ν(Φ)>0. This result is used to prove a more precise form of a theorem of Benson and Carlson on the existence of Scott components in the endomorphism ring of an indecomposable G-module, in case the module affords a G-invariant symmetric form.