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Eigenvalues of Finite Projective Planes with an Abelian Cartesian Group

机译：带有阿贝尔笛卡尔群的有限射影平面的特征值

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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|≡x~(n~2) 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.
机译：令M为n阶投影平面的入射矩阵。在Desarguesian情况下计算M的特征值，并在平面允许（P，L）-传递性G，| G |的假设下获得M的标准形式。 = n。对M的研究被简化为一个主子矩阵A，它是相关仿射平面的n〜2条线的入射矩阵。在这种情况下，A是Cayley置换表示R（G）的n阶广义Hadamard矩阵。在这些条件下，证明了当A的特征值是实数时，G是一个2族，n = 2〜r。如果G是阿贝尔阶，则特征多项式| xI- A |是n个多项式x —φ（A）|的乘积，φ是G的线性特征。该公式用于证明n是自然条件下在A和频谱（A）上的素数。可以推测，对于n的每个主除数p，| xI -A |≡x〜（n〜2）mod p表示该猜想的真值意味着n = | G |。是主要力量。

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