On models of the braid arrangement and their hidden symmetries

TL;DR

This paper investigates the symmetries of De Concini-Procesi models of the braid arrangement, revealing why only the minimal model admits an extended symmetric group action and constructing a supermaximal model with such symmetry.

Contribution

It introduces the supermaximal model that admits an extended $S_{n+1}$ action and provides an explicit basis for its integer cohomology, advancing understanding of hidden symmetries.

Findings

Only the minimal model admits the extended $S_{n+1}$ symmetry.

Constructed the supermaximal model with extended symmetry.

Described a basis for the integer cohomology of the supermaximal model.

Abstract

The De Concini-Procesi wonderful models of the braid arrangement of type $A_{n-1}$ are equipped with a natural $S_n$ action, but only the minimal model admits an `hidden' symmetry, i.e. an action of $S_{n+1}$ 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 $S_{n+1}$ 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 $S_{n+k}$ 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.