ABSTRACT
The classical indexation of Schubert cells in Grassmannians, by increasing functions from an integral interval into another, is not easily adapted to Flag varieties. This chapter introduces a more suitable general indexation by Relative Position Matrices for Schubert varieties in Flag varieties. These matrices are in bijection with classes of the symmetric group and thus give a geometric interpretation of the orbits in Bruhat decomposition of the linear group. The set of combinatorial flags may be identified with the parabolics subsets of the set of roots R(E), and a group theoretical description of Drap(E) is obtained. The chapter provides an example of the action of the Weyl group on the apartment of the building of a reductive group. It describes about the relationship between the parametrization by normalization of a Schubert cell, with that one given by a unipotent R-subgroup.
