Summary: We consider two natural embeddings between Artin groups: the group \(G_{\widetilde A_{n-1}}\) of type \(\widetilde A_{n-1}\) embeds into the group \(G_{B_n}\) of type \(B_n\); \(G_{B_n}\) in turn embeds into the classical braid group \(Br_{n+1}:=G_{A_n}\) of type \(A_n\). The cohomologies of these groups are related, by standard results, in a precise way. By using techniques developed in previous papers, we give precise formulas (sketching the proofs) for the cohomology of \(G_{B_n}\) with coefficients over the module \(\mathbb{Q}[q^{\pm 1},t^{\pm 1}]\), where the action is \((-q)\)-multiplication for the standard generators associated to the first \(n-1\) nodes of the Dynkin diagram, while is \((-t)\)-multiplication for the generator associated to the last node.
As a corollary we obtain the rational cohomology for \(G_{\widetilde A_n}\) as well as the cohomology of \(Br_{n+1}\) with coefficients in the \((n+1)\)-dimensional representation obtained by D. Tong, S. Yang and Z. Ma [Commun. Theor. Phys. 26, No. 4, 483-486 (1996; Zbl 1002.20500)].
We stress the topological significance, recalling some constructions of explicit finite CW-complexes for orbit spaces of Artin groups. In case of groups of infinite type, we indicate the (few) variations to be done with respect to the finite type case [see M. Salvetti, Math. Res. Lett. 1, No. 5, 565-577 (1994; Zbl 0847.55011)]. For affine groups, some of these orbit spaces are known to be \(K(\pi,1)\) spaces (in particular, for type \(\widetilde A_n\)).
We point out that the above cohomology of \(G_{B_n}\) gives (as a module over the monodromy operator) the rational cohomology of the fibre (analog to a Milnor fibre) of the natural fibration of \(K(G_{B_n},1)\) onto the 2-torus.
For the entire collection see [Zbl 1133.55001].