Title: Character tables and the problem of existence of finite projective planes,
Abstract: Recently, the authors of the present work (together with M. N. Kolountzakis)
introduced a new version of the non-commutative Delsarte scheme and applied it
to the problem of mutually unbiased bases. Here we use this method to
investigate the existence of a finite projective plane of a given order d. In
particular, a short new proof is obtained for the nonexistence of a projective
plane of order 6. For higher orders like 10 and 12, the method is non decisive
but could turn out to give important supplementary informations.