On models of the braid arrangement and their hidden symmetries

The De Concini-Procesi wonderful models of the braid arrangement of type An-1 are equipped with a natural Sn action, but only the minimal model admits an "hidden" symmetry, that is, an action of Sn+1 that comes from its moduli space interpretation. In this paper, we explain why the non minimal models do not admit this extended action: they are "too small". In particular, we construct a supermaximal model which is the smallest model that can be projected on to the maximal model and again admits an extended Sn+1 action. We give an explicit description of a basis for the integer cohomology of this supermaximal model. Furthermore, we deal with another hidden extended action of the symmetric group: we observe that the symmetric group Sn+k acts by permutation on the set of k-codimensional strata of the minimal model. Even if this happens at a purely combinatorial level, it gives rise to an interesting permutation action on the elements of a basis of the integer cohomology.