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Irreducible rank-three reflection-arrangement counts

PDF p. 28: §7.1, paragraph following Figure 1

Statement

An irreducible finite real reflection arrangement in dimension three has $6$ , $9$ or $15$ hyperplanes, corresponding respectively to types $A_{3}$ , $B_{3}$ and $H_{3}$ . Here a finite real reflection arrangement is the full arrangement of a finite linear reflection group, and irreducibility excludes nontrivial product decompositions, including an empty positive-dimensional factor, as in the local prerequisites.

Facts & Assumptions

[F1]

For an essential finite real reflection group, the reflections in the walls of a chamber form a Coxeter generating set of cardinality equal to the dimension (Chamber walls of an essential finite reflection group give full Coxeter rank). In the irreducible three-dimensional case its Coxeter diagram is connected (Irreducible reflection arrangements give connected Coxeter diagrams).

[F2]

A connected finite rank-three Coxeter system has type $A_{3}$ , $B_{3}$ or $H_{3}$ (Classification of finite Coxeter systems, including the H and dihedral families, part (1)).

[F3]

For an essential finite real reflection group with its chamber-wall Coxeter system, the arrangement hyperplanes biject with the Coxeter reflections (The hyperplanes are in bijection with Coxeter reflections).

[F4]

The reflection cardinalities for those three types are $6$ , $9$ and $15$ (Reflection cardinalities of the three connected finite rank-three types).

[F5]

The full reflection-arrangement and product-decomposition conventions, the invariant metric, and essentiality under irreducibility and $\dim E > 1$ are supplied by Invariant metric, product arrangements and essentiality.

Proof

Given: An irreducible finite real reflection arrangement $A (W)$ on a three-dimensional real space, with hyperplanes as in Linear hyperplane arrangements and local finiteness and reflection, product and irreducibility conventions as in Invariant metric, product arrangements and essentiality. Chambers, walls and the chamber-wall Coxeter system use Chamber walls of an essential finite reflection group give full Coxeter rank; Coxeter diagrams and their types use Coxeter-system, roots, length and weak-order conventions. Goal: Determine all possible types and hyperplane counts.

Proof technique: direct.

1.1givenF5F1F2

Since the given arrangement is irreducible and $\dim E = 3 > 1$ , [F5] supplies a $W$ -invariant positive-definite inner product and essentiality $E^{W} = 0$ . Thus [F1] applies: the chamber-wall Coxeter system has rank three and connected diagram. The underlying group is finite by the definition of a finite real reflection arrangement, so this system satisfies [F2]. Reading the rank-three members of the classification leaves precisely $A_{3}$ , $B_{3}$ and $H_{3}$ ; the other listed irreducible types have different ranks.

2.1step 1.1F3F4∎

Essentiality and the chamber-wall Coxeter system established in step 1.1 satisfy the hypotheses of [F3]. Therefore the number of original geometric hyperplanes equals the cardinality of the abstract reflection set of that Coxeter system. Applying [F4] in the three cases gives respectively $6$ , $9$ and $15$ . This proves the retained statement, with the rank, irreducibility and counting bridge supplied explicitly.

Depends on

Used by · 1

A relative arrangement need not be a reflection arrangement

Dependency tree · 2 levels

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF pp. 28; §7.1, paragraph following Figure 1