The mod \(2\) cohomology of the infinite families of Coxeter groups of type \(B\) and \(D\) as almost-Hopf rings (Q6188365)

The Coxeter groups of type \(W_{\mathrm{B}_{n}}\) and \(W_{\mathrm{D}_{n}}\) are two well-known infinite families of finite reflection groups. In the paper under review, the author describes a Hopf ring structure on the direct sum of the cohomology groups \(\bigoplus_{n \geq 0} H^{\ast}(W_{\mathrm{B}_{n}}; \mathbb{F}_{2})\) of the Coxeter groups of type \(W_{\mathrm{B}_{n}}\), and an almost-Hopf ring structure on the direct sum of the cohomology groups \(\bigoplus_{n \geq 0} H^{\ast}(W_{\mathrm{D}_{n}}; \mathbb{F}_{2})\) of the Coxeter groups of type \(W_{\mathrm{D}_{n}}\) with coefficients in the field with two elements \(\mathbb{F}_{2}\). He provides presentations with generators and relations, determines additive bases and compute the Steenrod algebra action. The generators are described both in terms of a geometric construction by \textit{C. De Concini} and \textit{M. Salvetti} [Math. Res. Lett. 7, No. 2--3, 213--232 (2000; Zbl 0972.20030)] and their restriction to elementary abelian \(2\)-subgroups.