Permanents and Lorentzian time-semidefinite matrices (Q1124900)

Permanents of Lorentzian time-semidefinite matrices defined as Hermitian matrices with nonnegative diagonal elements and with at most one positive eigenvalue are analyzed. Using a theorem of \textit{J. H. Grace} [Proc. Camb. Philos. Soc. 11, 352-357 (1900)] it is shown in terms of Hermitian symmetric forms on a finite dimensional complex vector space with a symmetric form of a Lorentz signature that such matrices have a nonnegative permanent. A generalization of the theorem of Grace due to \textit{L. Hörmander} [Math. Scand. 2, 55-64 (1954; Zbl 0058.25502)] is extended into a quantitative version for permanents. The inequalities and relations for Lorentzian time-semidefinite matrices are derived and applied to a Young tableau \(T\) of shape \(\lambda\) and other examples of matrix functions.