  We consider tangent cones of Schubert varieties in the complete flag variety,
and investigate the problem when the tangent cones of two different Schubert
varieties coincide. We give a sufficient condition for such coincidence, and
formulate a conjecture that provides a necessary condition. In particular, we
show that all Schubert varieties corresponding to the Coxeter elements of the
Weyl group have the same tangent cone. Our main tool is the notion of pillar
entries in the rank matrix counting the dimensions of the intersections of a
given flag with the standard one. This notion is a version of Fulton's
essential set. We calculate the dimension of a Schubert variety in terms of the
pillar entries of the rank matrix.
