Invariants in the cohomology of the complement of quaternionic reflection arrangements

TL;DR

This paper extends the study of invariants in the cohomology of hyperplane arrangement complements from complex reflection groups to quaternionic reflection groups, revealing new Poincaré polynomial families.

Contribution

It generalizes the invariants and Poincaré series results to quaternionic reflection groups, identifying a new class of Poincaré polynomials unique to the quaternionic case.

Findings

Generalization of Hilbert–Poincaré series to quaternionic groups

Discovery of a new family of Poincaré polynomials for certain quaternionic groups

Discussion of bases for G-invariant cohomology spaces

Abstract

Let $\mathcal A$ be a hyperplane arrangement in a vector space $V$ and $G \leq GL(V)$ a group fixing $\mathcal A$. In case when $G$ is a complex reflection group and $\mathcal A=\mathcal A(G)$ is its reflection arrangement in $V$, Douglass, Pfeiffer, and R\"ohrle studied the invariants of the $Q G$-module $H^*(M(\mathcal A);Q)$, the rational, singular cohomology of the complement space $M(\mathcal A)$ in $V$. In this paper we generalize the work in Douglass, Pfeiffer, and R\"ohrle to the case of quaternionic reflection groups. We obtain a straightforward generalization of the Hilbert--Poincar\'e series of the ring of invariants in the cohomology from the complex case when the quaternionic reflection group is complex-reducible according to Cohen's classification. Surprisingly, only one additional family of new types of Poincar\'e polynomials occurs in the quaternionic setting which is not realised in the complex case, namely those of a particular class of imprimitive irreducible quaternionic reflection groups. Finally, we discuss bases of the space of $G$-invariants in $H^*(M(\mathcal A);Q)$.