The mod 2 cohomology rings of the alternating subgroups of the Coxeter groups of Type B

TL;DR

This paper investigates the mod 2 cohomology rings of the alternating subgroups of Coxeter groups of Type B, revealing an almost-Hopf ring structure and providing a complete presentation using advanced algebraic techniques.

Contribution

It introduces a detailed algebraic structure for these cohomology rings and applies novel computational methods to fully describe their presentation.

Findings

The cohomology groups form an almost-Hopf ring structure.

A complete presentation of the cohomology ring is provided.

The techniques of Giusti and Sinha are effectively applied.

Abstract

We show that the direct sum of the cohomology groups of the alternating subgroups of the family of Coxeter groups of Type B exhibits an almost-Hopf ring structure. We apply techniques developed by Giusti and Sinha to fully compute a presentation of this structure for mod 2 coefficient.