  A characteristic polynomial is an important invariant in the field of
hyperplane arrangement. For the Linial arrangement of any irreducible root
system, Postnikov and Stanley conjectured that all roots of the characteristic
polynomial have the same real part. In relation to this conjecture, Yoshinaga
obtained an explicit relationship between the characteristic quasi-polynomial
and the Ehrhart quasi-polynomial for the fundamental alcove. In this paper, we
calculate Yoshinaga's explicit formula through the decomposition of the Ehrhart
quasi-polynomial into several quasi-polynomials and a modified shift operator,
and obtain new formulas for the characteristic quasi-polynomial of the Linial
arrangement. In particular, when the parameter of the Linial arrangement is
relatively prime to the period of the Ehrhart quasi-polynomial, we prove the
Postnikov-Stanley Linial arrangement conjecture. This generalizes some of the
results for the root systems of classical types that have been proved by
Postnikov-Stanley and Athanasiadis. For other cases, we verify this conjecture
for exceptional root systems using a computational approach.
