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The relative normaliser action need not be chamber-transitive

PDF p. 12: Remark 3.25

Statement

Let $(W, S)$ be a finite-rank Coxeter system, and fix $J \subseteq S$ . For $I \subseteq S$ , let $W_{I}$ be the subgroup generated by $I$ , and let $ℓ$ be word length in $S$ . A chamber label is a pair $(x, I)$ with $W_{J} x = x W_{I}$ for which $x$ has minimum length in $x W_{I}$ . The normaliser quotient $N (W, J) = {Norm}_{W} (W_{J}) / W_{J}$ acts on these labels, and hence on the corresponding relative chambers. An action is transitive if any chamber can be carried to any other by an acting element.

When $J = \emptyset$ , this action is transitive. In general it need not be transitive: for type $A_{2}$ , with $S = {s, t}$ , $m (s, t) = 3$ and $J = {s}$ , the labels $(1, {s})$ and $(t s, {t})$ lie in different orbits.

Statement references

Algebraic chamber-label set

Normalisers and their quotients

Real Coxeter representation and contragredient notation

Labelling theorem for relative chambers

Facts & Assumptions

[F1]

Labels correspond bijectively to relative chambers. The normaliser quotient preserves the second component of a label and acts transitively on all labels with a fixed second component. Labelling theorem for relative chambers Normaliser action on labels, orbits and adjacency

[F2]

In the canonical real representation, the simple roots $e_{u}$ form a basis, $u (e_{v}) = e_{v} - 2 B (e_{v}, e_{u}) e_{u}$ , and $B (e_{u}, e_{v}) = - \cos (π / m (u, v))$ for distinct generators with finite label. Each simple reflection is nonidentity. Real Coxeter representation and contragredient notation

[F3]

A root is positive when its simple-root coordinates are nonnegative. For $w \in W$ and $u \in S$ , $ℓ (w u) > ℓ (w)$ when $w e_{u}$ is positive. Root signs, inversions and descent translation

[F4]

The Coxeter presentation has $s^{2} = t^{2} = 1$ and, when $m (s, t) = 3$ , $s t s = t s t$ . The label and quotient conventions are those in the Statement. Real Coxeter representation and contragredient notation Algebraic chamber-label set Normalisers and their quotients

Proof

Given: The finite-rank Coxeter system and labels in the Statement.

Given: The data in the Statement.

Goal: Prove transitivity for empty $J$ and verify the stated nontransitive example.

1.1F1F2F4

If $J = \emptyset$ , then $W_{J} = {1}$ . For any label $(x, I)$ , its defining equality gives $x W_{I} = {x}$ , hence $W_{I} = {1}$ . Each simple generator is nonidentity by [F2], so $I = \emptyset$ . All labels therefore have the same second component. By [F1], the action is transitive on labels and hence on chambers.

1.2F2F3F4

In type $A_{2}$ , put $J = {s}$ and $x = t s$ . The pair $(1, {s})$ is a label: $W_{s} 1 = 1 W_{s}$ , and $1$ has length zero. The braid relation gives $s x = s t s = t s t = x t$ , so $W_{s} x = x W_{t}$ . Since $B (e_{s}, e_{t}) = - 1 / 2$ , the reflection formula gives $s e_{t} = e_{t} + e_{s}$ and $t (s e_{t}) = e_{s}$ . Thus $x e_{t} = e_{s}$ is positive, and [F3] gives $ℓ (x t) > ℓ (x)$ . As $W_{t} = {1, t}$ , $x$ is the minimum in $x W_{t}$ , so $(x, {t})$ is also a label.

2.1F1step 1.2∎

The two labels have different second components. By [F1], the action preserves that component, so they cannot lie in the same orbit. Their corresponding chambers are distinct by the chamber bijection, proving the claimed failure of transitivity.

Source notes

The original article attributes the general observation and points to Example 3.12 and Section 7. The rank-two calculation above is a local argument and does not use that observation as a premise.

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 12, Remark 3.25