Character tables and the problem of existence of finite projective planes

TL;DR

This paper applies a novel non-commutative Delsarte scheme to study the existence of finite projective planes, providing a new proof for the nonexistence of a plane of order 6 and offering insights for higher orders.

Contribution

It introduces a new application of the non-commutative Delsarte scheme to finite projective planes, including a simplified proof for the nonexistence of order 6.

Findings

Proves the nonexistence of a projective plane of order 6.

Method is inconclusive for orders 10 and 12 but may provide additional information.

Demonstrates the utility of the non-commutative Delsarte scheme in combinatorial geometry.

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.