Cohomology of the hyperplane complement of a quaternionic reflection group

TL;DR

This paper calculates the cohomology of hyperplane complements related to quaternionic reflection groups, revealing a factorization pattern with mostly linear factors and some quadratic ones in special cases.

Contribution

It provides the first detailed computation of the graded cohomology for these quaternionic reflection group complements, highlighting their factorization properties.

Findings

Cohomology factors into irreducible components with positive integer coefficients.

Most factors are linear, with at most one quadratic factor in exceptional cases.

The results apply specifically to the hyperplane complements of quaternionic reflection groups.

Abstract

We compute the graded rank of the cohomology of the hyperplane complement associated with a quaternionic reflection group, and observe that it factors into irreducible factors with positive integer coefficients. For an irreducible group, these irreducible factors are all linear except at most one irreducible quadratic factor, which occurs for precisely three of the exceptional groups.