Cohomology of the hyperplane complement of a quaternionic reflection group

TL;DR

This work computes the cohomology of hyperplane complements for quaternionic reflection groups by deriving the Poincaré polynomial $p_W(t)$ and codimension polynomial $c_W(t)$ across all irreducible quaternionic types. It leverages Orlik–Solomon theory, line-system reformulations, and Cohen’s Coh classification to obtain explicit product formulas for the infinite family $W_n( extGamma, extDelta)$ and detailed flat enumerations for the exceptional groups, yielding factorization patterns of $p_W(t)$ into mostly linear factors with at most one quadratic factor in three exceptional cases. The analysis highlights a striking parallel with complex reflection groups: the polynomials factor with positive integer coefficients, suggesting deeper structural commonalities, and furnishes rich data (μ and e invariants) that illuminate parabolic subgroups and potential links to symplectic reflection algebras. The results provide concrete, computable invariants for a broad class of quaternionic reflection groups and pave the way for further algebraic and topological investigations into parity, symmetry, and invariant theory in quaternionic settings.

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.

Theorems & Definitions (37)

• Theorem 1.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• Lemma 3.4
• proof
• Lemma 5.1
• proof