The mod 2 cohomology of the infinite families of Coxeter groups of type B and D as almost Hopf rings

TL;DR

This paper constructs and analyzes algebraic structures on the mod 2 cohomology of Coxeter groups of types B and D, providing generators, relations, bases, and Steenrod algebra actions.

Contribution

It introduces a Hopf ring structure for type B and an almost-Hopf ring for type D cohomology, with explicit presentations and geometric interpretations.

Findings

Established Hopf and almost-Hopf ring structures

Provided generators, relations, and bases for cohomology groups

Computed Steenrod algebra actions on the cohomology

Abstract

We describe a Hopf ring structure on the direct sum of the cohomology groups $\bigoplus_{n \geq 0} H^* \left( W_{B_n}; \mathbb{F}_2 \right)$ of the Coxeter groups of type $B_n$, and an almost-Hopf ring structure on the direct sum of the cohomology groups $\bigoplus_{n \geq 0} H^* \left( W_{D_n}; \mathbb{F}_2 \right)$ of the Coxeter groups of type $D_n$, with coefficient in the field with two elements $\mathbb{F}_2$. We give presentations with generators and relations, determine additive bases and compute the Steenrod algebra action. The generators are described both in terms of a geometric construction by De Concini and Salvetti and in terms of their restriction to elementary abelian 2-subgroups.