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Normaliser action on labels, orbits and adjacency

PDF p. 12: Lemma 3.24

Statement

Let $(W, S)$ be a finite-rank Coxeter system with vertex set $Γ_{0}$ and fix $J \subseteq Γ_{0}$ . For $I \subseteq Γ_{0}$ , write $W_{I}$ for the standard parabolic subgroup generated by the simple reflections indexed by $I$ , and let $ℓ$ be Coxeter length. The chamber-label set $C h a m (J)$ consists of the pairs $(x, I)$ such that $W_{J} x = x W_{I}$ and $x$ has minimum length in $x W_{I}$ . Write $N (W, J) = {Norm}_{W} (W_{J}) / W_{J}$ for the normaliser quotient.

Under the chamber bijection $(x, I) \mapsto x C_{I}$ , the quotient action on the relative cone induces an action on $C h a m (J)$ . For $\overset{ˉ}{g} \in N (W, J)$ and any representative $g \in {Norm}_{W} (W_{J})$ , it is given by $\overset{ˉ}{g} \cdot (x, I) = (y, I)$ , where $y$ is the unique minimum-length element of $g x W_{I}$ . Any two labels with the same second component lie in the same orbit, and the action is free: a quotient element fixing a label is the identity.

If $ω_{a, I} (x, I) = (x v_{a, I}, K)$ is a simple wall crossing to an adjacent chamber, then $ω_{a, I} (\overset{ˉ}{g} \cdot (x, I)) = \overset{ˉ}{g} \cdot ω_{a, I} (x, I)$ . Thus the action preserves adjacency and the crossing index $a$ .

Statement references

Algebraic chamber-label set

Normalisers and their quotients

Parabolic coordinate subspaces and fundamental faces

Simple wall-crossing elements and maps

Coxeter-system, roots, length and weak-order conventions

Tits cone intersections, relative roots and arrangements

Chambers, walls and adjacency

Facts & Assumptions

[F1]

A label $(x, I)$ means $W_{J} x = x W_{I}$ and $x$ is the minimum of $x W_{I}$ . The quotient is formed from the normaliser by its normal subgroup $W_{J}$ . Algebraic chamber-label set Normalisers and their quotients

[F2]

The map $(x, I) \mapsto x C_{I}$ is a bijection from the label set to the relative chambers. The quotient acts on the relative cone and sends $x C_{I}$ to $(g x) C_{I}$ for any normaliser lift $g$ . Labelling theorem for relative chambers Faithful normaliser-quotient action on the relative cone

[F3]

Every right coset of a standard parabolic has a unique minimum-length representative. Every element of $W_{I}$ fixes each point of the fundamental face $C_{I}$ . Parabolic root subsystems and unique coset minima Stabilizer of a fundamental parabolic face Parabolic coordinate subspaces and fundamental faces

[F4]

For an already adjacent pair with first label $(x, I)$ and wall index $a \notin I$ , the adjacent-wall proposition gives the finite component required by the crossing definition and identifies the other label as $(x v_{a, I}, K)$ . The element $v_{a, I}$ and subset $K$ depend only on $I$ and $a$ . Adjacent chambers and inverse wall-crossing elements Simple wall-crossing elements and maps

[F5]

The roots are $Φ = {w α_{i} : w \in W, i \in Γ_{0}}$ , and the dual action is $(g f) (v) = f (g^{- 1} v)$ . Write $E = Θ_{J}$ and $T = C o n e (J)^{\circ}$ , with interior taken in the coordinate Euclidean topology on $E$ . The relative arrangement consists of the distinct proper traces $H_{α} \cap E$ , where $H_{α} = {f \in Θ : f (α) = 0}$ ; traces missing $T$ are omitted when restricting to that domain. Coxeter-system, roots, length and weak-order conventions Tits cone intersections, relative roots and arrangements Labelling theorem for relative chambers

[F6]

A wall is an intersection of a chamber closure with an arrangement hyperplane whose affine dimension is one less than that of the chamber. Two distinct chambers are adjacent when they share a wall. Closures for the relative arrangement are taken in $E$ . Chambers, walls and adjacency

[F7]

If the component required by the definition of $ω_{a, I}$ is finite, then for every label $(x, I)$ the pair $(x v_{a, I}, K)$ is itself a chamber label. This label assertion is supplied independently of the normaliser action. A finite connected component gives the opposite relative chamber Simple wall-crossing elements and maps

Proof

Given: The finite-rank Coxeter system, arbitrary subset $J$ , chamber label and quotient element in the statement. No spherical-type hypothesis is assumed.

Given: A label $(x, I)$ and a normaliser representative $g$ as above.

Goal: Establish the action formula, its orbit and freeness assertions, and its compatibility with simple crossings.

1.1F1F2F3

Let $y$ be the unique minimum of $g x W_{I}$ , so $g x = y u$ for some $u \in W_{I}$ . Since $u$ fixes $C_{I}$ pointwise, $(g x) C_{I} = y C_{I}$ . Moreover $W_{J} g x = g W_{J} x = g x W_{I} = y W_{I}$ , because $g$ normalises $W_{J}$ . Thus $(y, I)$ is a label. By the chamber bijection and the quotient action on the cone, it is the label of the image chamber. This gives the formula and proves that it is independent of the representative $g$ ; the action laws follow from the quotient action on chambers.

2.1F2F5F6step 1.1

By [F2], the restriction of $g$ is an invertible linear map of the finite-dimensional space $E$ , hence a homeomorphism, and it preserves $C o n e (J)$ and therefore its interior $T$ . By the root-orbit definition in [F5], $g Φ = Φ$ . The dual-action formula gives $(g f) (g α) = f (α)$ , so $g H_{α} = H_{g α}$ . Since $g E = E$ , it follows that $g (H_{α} \cap E) = H_{g α} \cap E$ ; properness of the trace and whether it meets $T$ are preserved. Thus $g$ permutes the relative arrangement on $T$ . If chambers $D, D^{'}$ share a wall $F = \overline{D} \cap H = \overline{D^{'}} \cap H^{'}$ with supporting relative hyperplanes $H, H^{'}$ , then $g F = \overline{g D} \cap g H = \overline{g D^{'}} \cap g H^{'}$ , because a homeomorphism commutes with closure. An invertible linear map preserves affine dimension, and it keeps the two chambers distinct. Hence $g F$ is a wall of both image chambers, proving preservation of adjacency without invoking any forward crossing assertion.

2.2F1F3step 1.1

If $(x, I)$ and $(z, I)$ are labels, set $g = z x^{- 1}$ . Their defining equalities give $g W_{J} g^{- 1} = z (x^{- 1} W_{J} x) z^{- 1} = z W_{I} z^{- 1} = W_{J}$ . Thus $g$ is in the normaliser, and $g x W_{I} = z W_{I}$ has minimum $z$ . Step 1.1 sends $(x, I)$ to $(z, I)$ , proving transitivity within each fixed second component.

2.3F1step 1.1

If $\overset{ˉ}{g}$ fixes $(x, I)$ , step 1.1 gives $g x W_{I} = x W_{I}$ , hence $g \in x W_{I} x^{- 1} = W_{J}$ . Its quotient class is therefore the identity. This proves freeness.

2.4F1F4step 1.1

Suppose the crossing from $(x, I)$ has target $(x v, K)$ , where $v = v_{a, I}$ . Both are labels, so $x^{- 1} W_{J} x = W_{I}$ and $(x v)^{- 1} W_{J} (x v) = W_{K}$ ; consequently $v^{- 1} W_{I} v = W_{K}$ . Write $g x = y u$ as in step 1.1. Then $g x v = y v (v^{- 1} u v)$ and $v^{- 1} u v \in W_{K}$ , so $g x v W_{K} = y v W_{K}$ .

3.1F1F3F4F7step 1.1step 2.1step 2.4∎

The original adjacent pair in step 2.4 supplies the finite component for $I, a$ by [F4]. The image label $(y, I)$ has the same $I$ , so the same finiteness condition holds and the crossing definition uses the same $v = v_{a, I}$ and $K$ . By [F7], $(y v, K)$ is a label; in particular $y v$ is the minimum of $y v W_{K}$ . Step 2.4 identifies that coset with $g x v W_{K}$ . The action formula in step 1.1 and uniqueness of its minimum therefore show that the image of $(x v, K)$ is exactly $(y v, K)$ . Step 2.1 independently shows that this image chamber is adjacent to the chamber labelled $(y, I)$ . Thus the crossing at index $a$ from the image label has precisely this adjacent target, and $ω_{a, I} (\overset{ˉ}{g} \cdot (x, I)) = \overset{ˉ}{g} \cdot ω_{a, I} (x, I)$ . Every adjacent pair has a crossing description by [F4], completing all the assertions.

Source notes

The original article supplies attribution for the statement. The proof above uses the declared mathematical suppliers and the local arrangement-preservation argument; it does not use the article's assertion as a premise.

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 12, Lemma 3.24