On Models of the Braid Arrangement and their Hidden Symmetries

Abstract

The De Concini-Procesi wonderful models of the braid arrangement of type A n−1 are equipped with a natural Sn 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 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.

Nested sets, set partitions and Kirkman–Cayley dissection numbers

“…As one can easily check, when k > 3 this action is not compatible with the usual S n action. A geometric application of this remark will be discussed in the paper [2].…”

Projective wonderful models for toric arrangements

“…(2) N = ∅, by Lemma 4.2 we can find a σ ∈ Θ (2) N such that the triple (M σ Θ , ε σ Θ , q σ Θ ) is smaller giving a contradiction. This settles the case Ξ = {χ}…”

“…Suppose ε σ = 1, then χ, e 2 = −M σ , hence χ, e 1 + e 2 = 0 so that all these cones do not lie in σ ∆ We deduce that P σ ∆ takes values which are at most equal to (M σ , σ ). Furthermore if τ ∈ σ ∆ (2) N is such that P σ ∆ (τ ) = (M σ , σ ), necessarily τ ∈ ∆ (2) N \ {σ} and everything follows. Let us now denote by M ∆ the family of fans which are obtained from ∆ by a repeated application of the following procedure: given a fan R, choose a two dimensional cone σ in R and create the new fan σ R. Theorem 4.1 (see also [9]).…”

Exponential formulas for models of complex reflection groups

“…Namely, the symmetric group S n+k−1 acts on the set of the F G(1,1,n) -nested sets S such that |S| = k and S includes {1, 2, ..., n}. These actions can be extended to the basis of cohomology described in Theorem 2.2: in [4] (Theorem 10.1) it was shown that one can obtain, by counting the orbits of these actions, an exponential (not recursive) formula for a series that computes the Betti numbers of the models Y G(1,1,n) .…”