Maximal determinants of matrices over the roots of unity

Abstract

We study the maximum absolute value of the determinant of matrices with entries in the set of ℓ-th roots of unity — this is a generalization of D-optimal designs and Hadamard's maximal determinant problem, which involves ±1 matrices. For general values of ℓ, we give sharpened determinantal upper bounds and constructions of matrices of large determinant. The maximal determinant problem in the cases $\ell =3$, $\ell =4$ is similar to the classical Hadamard maximal determinant problem for matrices with entries ±1, and many techniques can be generalized. For $\ell =3$ we give an additional construction of matrices with large determinant, and calculate the value of the maximal determinant of matrices with entries in the third-roots of unity for all orders $n<14$. Additionally, we survey the case $\ell =4$ and exhibit an infinite family of maximal determinant matrices of odd order over the fourth roots of unity.