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Normaliser-action wording in the D4 exposition

PDF p. 29: §7.2.1, paragraph beginning “By Theorem 4.7”

Statement

In the $D_{4}$ example, vertex $2$ is central, $J = {2}$ , and $W_{J} = ⟨ s_{2} ⟩$ . The normaliser is ${Norm}_{W} (W_{J}) = {g \in W : g W_{J} g^{- 1} = W_{J}}$ and $N (W, J) = {Norm}_{W} (W_{J}) / W_{J}$ . A relative chamber has a unique label $(x, I)$ satisfying $W_{J} x = x W_{I}$ and with $x$ of minimum length in $x W_{I}$ ; it is the face translate $x C_{I}$ .

The induced action of $N (W, J)$ on these chambers is free: a quotient element fixing a chamber is the identity. Two chambers are in the same orbit, meaning that one is the image of the other under a quotient element, exactly when their second labels agree. There are four orbits, indexed by $I = {1}, {2}, {3}, {4}$ , each containing eight chambers.

The article's paragraph instead says that the action of each $g \in {Norm}_{W} (W_{J})$ is free. The quotient formulation above makes the intended freeness precise: the nonidentity element $s_{2}$ acts trivially on every relative chamber.

Statement references

D4 diagram and generator conventions

Normalisers and their quotients

Algebraic chamber-label set

Labelling theorem for relative chambers

Faithful normaliser-quotient action on the relative cone

D4 chamber labels and complete adjacency graph

Facts & Assumptions

[F1]

The quotient action is induced by the normaliser action on the relative dual space; its kernel before quotienting is $W_{J}$ , and a lift $g$ sends $x C_{I}$ to $(g x) C_{I}$ . Faithful normaliser-quotient action on the relative cone Normalisers and their quotients

[F2]

Labels $(x, I)$ satisfy $x W_{I} x^{- 1} = W_{J}$ and correspond bijectively to chambers $x C_{I}$ . Algebraic chamber-label set Labelling theorem for relative chambers

[F3]

Each orbit in the Tits cone meets the closed fundamental chamber in exactly one point, and the stabiliser of every point of $C_{I}$ is $W_{I}$ . Tits cone geometry with its exact interior boundary

[F4]

In the completed $D_{4}$ chamber list, each of the singleton second labels $1, 2, 3, 4$ occurs exactly eight times and there are no other labels. D4 diagram and generator conventions D4 chamber labels and complete adjacency graph

Proof

Given: The $D_{4}$ data and $J$ in the Statement. Write $C_{I}$ for the fundamental face with zero simple-root coordinates exactly at $I$ .

Proof technique: direct.

Goal: Establish freeness, the orbit criterion and the stated correction to the normaliser wording.

1.1F1F2F3given

Suppose a normaliser element $g$ sends the chamber $x C_{I}$ to $y C_{K}$ . Choose $f \in C_{I}$ . Then $h f \in C_{K}$ , where $h = y^{- 1} g x$ . Both $f$ and $h f$ lie in the closed fundamental chamber and in the same orbit. By [F3], $h f = f$ ; their zero coordinates therefore give $I = K$ , and [F3] also gives $h \in W_{I}$ . In particular, if $y = x$ and $K = I$ , then $g \in x W_{I} x^{- 1} = W_{J}$ by [F2]. Hence a quotient element fixing a chamber is the identity, proving freeness.

2.1F1F2F4step 1.1

Conversely, let $(x, I)$ and $(y, I)$ be labels with the same second component. By [F2], both $x W_{I} x^{- 1}$ and $y W_{I} y^{- 1}$ equal $W_{J}$ . Thus $g = y x^{- 1}$ normalises $W_{J}$ , and [F1] sends $x C_{I}$ to $y C_{I}$ . Together with step 1.1, this proves the orbit criterion. Counting the labels in [F4] gives four orbits of eight chambers.

3.1F1F4step 1.1∎

The reflection formula in the $D_{4}$ data gives $s_{2} α_{2} = - α_{2}$ , so $s_{2} \neq 1$ . It belongs to $W_{J}$ and hence acts trivially on the relative space and its chambers by [F1]. Therefore the normaliser action itself is not free, whereas its quotient action is free by step 1.1. This completes the clarification.

Source notes

The article reference attributes the wording in its $D_{4}$ discussion. The proof uses the listed completed suppliers and does not take that wording as a premise.

Depends on

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 29, §7.2.1, paragraph beginning ‘By Theorem 4.7’