On Matrices Arising in Finite Field Hypergeometric Functions

Lehmer constructs four classes of matrices constructed from roots of unity for which the characteristic polynomials and the $k$-th powers can be determined explicitly. Here we study a class of matrices which arise naturally in transformation formulas of finite field hypergeometric functions and whose entries are roots of unity and zeroes. We determine the characteristic polynomial, eigenvalues, eigenvectors, and $k$-th powers of these matrices. In particular, the eigenvalues are natural systems of products of Jacobi sums.