The S n+1 Action on Spherical Models and Supermaximal Models of Tipe A n−1

Abstract

In this paper we recall the construction of the De Concini–Procesi wonderful models of the braid arrangement: these models, in the case of the braid arrangement of type An−1, are equipped with a natural S n action, but only the minimal model admits an ‘hidden’ symmetry, i.e. an action of Sn+1 that comes from its moduli space interpretation. We show that this hidden action can be lifted to the face poset of the corresponding minimal real spherical model and we compute the number of its orbits. Then we provide a spherical version of the construction of the supermaximal model (see Callegaro, Gaiffi, On models of the braid arrangement and their hidden symmetries. Int. Math. Res. Not. (published online 2015). doi: 10.1093/imrn/rnv009), i.e. the smallest model that can be projected onto the maximal model and again admits the extended Sn+1 action.