The Miracle of Integer Eigenvalues

For partially ordered sets \((X, \preccurlyeq)\) , we consider the square matrices \(M^{X}\) with rows and columns indexed by linear extensions of the partial order on \(X\) . Each entry \((M^{X})_{PQ}\) is a formal variable defined by a pedestal of the linear order \(Q\) with respect to linear order \(P\) . We show that all eigenvalues of any such matrix \(M^{X}\) are \(\mathbb{Z}\) -linear combinations of those variables.