Incidence matrices of finite projective planes and their eigenvalues
The author provides two results concerning the incidence matrix of a finite projective plane. Let \(A\) be the 0-1 incidence matrix of a projective plane of order \(n\). \(A\) will be viewed as a square matrix of order \(N\) (\(=n^2+n+1\)) over \(\mathbb{C}\), the complex numbers. Set \(f_A(x)=\text{det}(xI-A)\), and let \(F_A\) be the splitting field of \(f_A(x)\) with \(F_A\subseteq\mathbb{C}\), and set \(G_A=\text{Gal}(F_A/\mathbb{Q})\). Let \(c_A\) be that element of \(G_A\) determined by complex conjugation, so that \(c_A(\alpha)=\overline\alpha\) for every zero \(\alpha\) of \(f_A(x)\) in \(\mathbb{C}\). We say that \(A\) splits if and only if \(c_A\) is of order 2, and \(\langle c_A\rangle\) is a direct factor of \(G_A\). Let \({\mathcal T}_N\) denote the set of all permutation matrices of \(\text{GL} (N,\mathbb{C})\) which represent transpositions. The first result is: Theorem 1. If \(n\) is not a square and \(A\) is not split, then \(AT\) is split for all \(T\in{\mathcal T}_N.\) For any incidence matrix \(A\), write \(f_A(x)=(x-(n+1))g_A(x)\). Then \(g_A(x)\) is a monic polynomial with integer coefficients such that \(g_A(n+1)\neq 0\). The second result is the following interlacing theorem: Theorem 2. Suppose \(n\) is not a square and \(T\in{\mathcal T}_N\). Let \(g_A(x)=D(x)E(x)\) and \(g_{AT}(x)=D(x)F(x)\), where \(D(x)\) is the greatest common divisor of \(g_A(x)\) and \(g_{AT}(x)\). Then \(E(x)\) and \(F(x)\) are square free, and on the circle of radius \(\sqrt n\) centered at the origin, between any two successive zeroes of \(E(x)\) there is precisely one zero of \(F(x)\). The author remarks that with some mild modifications, the above results can be adapted to the case where \(n\) is a square.
scientific article; zbMATH DE number 1019696