  We study the rational permutation braids, that is the elements of an
Artin-Tits group of spherical type which can be written $x^{-1} y$ where $x$
and $y$ are prefixes of the Garside element of the braid monoid. We give a
geometric characterization of these braids in type $A_n$ and $B_n$ and then
show that in spherical types different from $D_n$ the simple elements of the
dual braid monoid (for arbitrary choice of Coxeter element) embedded in the
braid group are rational permutation braids (we conjecture this to hold also in
type $D_n$).This property implies positivity properties of the polynomials
arising in the linear expansion of their images in the Iwahori-Hecke algebra
when expressed in the Kazhdan-Lusztig basis. In type $A_n$, it implies
positivity properties of their images in the Temperley-Lieb algebra when
expressed in the diagram basis.
