The Flag Complex

The Flag Complex is the Tits Building of the general linear group GL(k^r+1) so named by Mumford. The term Flag Complex refers to its underlying abstract simplicial complex, i.e. its combinatorial structure. This chapter introduces the building setting in the case of the linear group, and discusses a family of configurations varieties, which give rise to smooth resolutions of Schubert varieties, in terms of the building geometry introduced by Tits. The set of flags of k^r+1 adapted to the canonical basis, i.e. the combinatorial flags, is endowed with the structure of a simplicial complex, namely the first barycentric subdivision of the combinatorial (r + 1)-simplex. The natural action of on the combinatorial flags induces an isomorphism between these two complexes. The set of flags Drap(k^r+1) of k^r+1 endowed with a simplicial complex structure forms the Flag complex that contains the complex of adapted flags as a subcomplex. The set of combinatorial flags is naturally endowed with a building structure.