2014/06/05 by Filippo Callegaro, Callegaro, Filippo, Giovanni Gaiffi +1
Mathematics · #14N20 (Primary) #20C30 #57N65 (Secondary) #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT) #math.AT #math.CO #math.RT #msc:14N20 #msc:20C30 #msc:57N65
paper · pdf · doi:10.48550/arxiv.1406.1304
arxiv created 2014/06/05 · arxiv updated 2014/06/06
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, i.e. an action of Sn+1 that comes from its moduli space interpretation. In this paper we explain why the non minimal models don't admit this extended action: they are `too small'. In particular we construct a \em supermaximal model which is the smallest model that can be projected onto 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-codimensionl 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.