The maximal determinant of order 15 over the third roots of unity

Abstract

We prove that the largest absolute determinant of a $15\times15$ matrix with entries in $\{1,ω,ω^2\}$, where $ω^3=1$ and $ω\ne1$, is $120932352\sqrt{19}$. A matrix attaining this value was already known; the issue is maximality. The proof follows the Gram-matrix search method used in the real maximal-determinant problem. A sharp determinant bound in terms of the off-diagonal Gram energy, combined with Eisenstein congruences and a published classification of difference matrices, reduces a strict improvement to 23 possible energies. Componentwise Schur bounds and exact support catalogues then leave a small number of cases, excluded by orthogonality, rank, and spectral-intertwining conditions. We present the finite tests in a common framework and supply their replay programs. For the displayed extremizer we give a dephased representative, its Gram matrix and spectrum, and an exact computation of its monomial automorphism group, which has order 1008. The proof determines the value of the maximum, but does not classify all equality cases.