The extended action of Sn+1 on the cohomology algebra of the pure braid space

We start defining one of the main objects we want to study. Definition 1. The pure braid space (on n strings) is defined as Mn = {(z1,..., zn) ∈ Cn: zi 6 = zj ∀i 6 = j} As the name may suggest the space Mn has relevant connection with the theory of braid groups; indeed it is a classifying space for the pure braid group on n strings. We can track the beginning of our story back to [Arn69] where Arnol’d computed the cohomology H∗(Mn;C). In particular he proved that it is isomorphic to the skew-commutative algebra generated by elements {Ai,j: 1 ≤ i < j ≤ n} with relations Ai,jAi,k − Ai,jAj,k + Ai,kAj,k = 0 ∀i < j < k. (1) After [Arn69] the space Mn was studied in detail and many generalization arose. We can point out two important directions in which the results about Mn can be generalized: (i) Configuration spaces. Let M be a (topological, smooth, complex, etc..) mani-fold; the configuration space of n points in M is defined as Fn(M) = {(p1,..., pn) ∈Mn: pi 6 = pj ∀i 6 = j}. (ii) Hyperplane arrangement. Let V be a vector space over a field K, an hy-perplane arrangement on V is a finite collection of affine hyperplanes of V, A = {H1,..., Hn}. 1 The configuration space Fn(Rd) As the name may suggest the study of the spaces Fn(M) has relevant applications to mechanics. The study of Fn(M) focuses on its topology. Here we are concerned with the configuration space of n points in Rd. The cohomology algebra for Fn(Rd) (with complex coefficients) was first computed by Cohen et al. in [CLM76]. They 1 proved that it is isomorphic to the associative graded algebra generated by elements {Ai,j: 1 ≤ i, j ≤ n} of degree d − 1 with relations Ai,j = (−1)dAj,i (2) Ai,jAh,k = (−1)d−1Ah,kAi,j (3) Ai,jAi,k = Ak,j(Ai,k − Ai,j) ∀i, j ≤ k (4) It is easily seen that for the case d = 2 (where we have an homeomorphism Fn(R2) ∼= Mn) this presentation agrees with the Arnol’d one. Though we will not prove this result it may be worth commenting relations (2)-(4). The isomorphism between H∗(Fn(Rd);C) and this algebra can be described as follows Ai,j 7 → α∗i,j(γ) ∈ H∗(Fn(Rd);C) where αi,j: Fn(Rd) → Rd\{0} is the restriction of the linear ma