Eigenvalues of Finite Projective Planes with an Abelian Cartesian Group

Abstract

Let M be an incidence matrix for a projective plane of order n. The eigenvalues of M are calculated in the Desarguesian case and a standard form for M is obtained under the hypothesis that the plane admits a (P,L)-transitivity G, |G| = n. The study of M is reduced to a principal submatrix A which is an incidence matrix for n ² lines of an associated affine plane. In this case, A is a generalized Hadamard matrix of order n for the Cayley permutation representation R(G). Under these conditions it is shown that G is a 2-group and n = 2^r when the eigenvalues of A are real. If G is abelian, the characteristic polynomial |xI − A| is the product of the n polynomials |x − φ (A)|, φ a linear character of G. This formula is used to prove n is a prime power under natural conditions on A and spectrum(A). It is conjectured that |xI − A| ≡ xⁿ² mod p for each prime divisor p of n and the truth of the conjecture is shown to imply n = |G| is a prime power.