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Eight-element D4 normaliser quotient vertex group

PDF p. 29: §7.2.1 after Figure 4; red labels in Figure 3; PDF p. 30: §7.2.1 after Figure 4; red labels in Figure 3

Example

For any finite-rank Coxeter system in the conventions of Correspondence between chamber labels and Brink–Howlett morphisms, an arrow $x : J \to I$ of the Brink–Howlett groupoid $B H (J)$ lifts uniquely from the fundamental label $(e, J)$ to the label $(x, I)$ .

For the $D_{4}$ Coxeter diagram with central vertex $2$ , set $J = {2}$ and write $s = s_{1}$ , $t = s_{2}$ , $u = s_{3}$ , $v = s_{4}$ , and $e = 1$ . Abbreviate $(x, {2})$ to $(x, 2)$ . Let $W_{J} = ⟨ t ⟩$ , let ${Norm}_{W} (W_{J}) = {w \in W : w W_{J} w^{- 1} = W_{J}}$ , and put $N (W, J) = {Norm}_{W} (W_{J}) / W_{J}$ . Then the vertex group ${B H}_{J} = {Hom}_{B H (J)} (J, J)$ is a subgroup of $W$ , and the quotient map induces ${B H}_{J} ≅ N (W, J)$ . More precisely,

${B H}_{J} = {x : (x, 2) \in C h a m (J)} = {e, s t u t s, s t u v t s, s t v t s, v t s u t v, s t u v t s u t v t u, v t u t v, u t s v t u} .$

These are eight distinct group elements. The eight corresponding labels are the red vertices in the article's figure.

Example references

D4 diagram and generator conventions

Algebraic chamber-label set

Brink–Howlett groupoid

Correspondence between chamber labels and Brink–Howlett morphisms

Root-permuting normaliser complement

Coxeter normaliser semidirect product

D4 chamber labels and complete adjacency graph

Facts & Assumptions

[F1]

An associate of $J$ is a subset whose standard parabolic subgroup is conjugate to $W_{J}$ . The objects of $B H (J)$ are the associates of $J$ ; an arrow $x : I \to K$ is an element carrying the set of simple roots indexed by $K$ onto that indexed by $I$ . Products of arrow labels compose from left to right. Brink–Howlett groupoid

[F2]

The label set $C h a m (J)$ consists of pairs $(x, I)$ with $W_{J} x = x W_{I}$ and $x$ of minimum length in the right coset $x W_{I}$ . Its universal wall groupoid has exactly one arrow from $(x, I)$ to $(y, K)$ , labelled $x^{- 1} y$ . If $w : I \to K$ is a Brink–Howlett arrow and $(x, I)$ is a label, its unique lift from $(x, I)$ ends at $(x w, K)$ . Conversely, the label of an arrow between two such labels is a Brink–Howlett arrow. Algebraic chamber-label set Correspondence between chamber labels and Brink–Howlett morphisms

[F3]

The vertex group ${B H}_{J}$ is the root-permuting normaliser subgroup $N_{J}$ , and the quotient map restricts to an isomorphism $N_{J} \to N (W, J)$ . Root-permuting normaliser complement Coxeter normaliser semidirect product

[F4]

In the specified $D_{4}$ example the completed chamber table lists all 32 distinct labels. Exactly its vertices 1, 5, 7, 9, 24, 26, 28 and 32 have second component $2$ , with first components respectively $s t u t s, s t u v t s, e, s t v t s, v t s u t v, s t u v t s u t v t u, v t u t v, u t s v t u$ . D4 diagram and generator conventions D4 chamber labels and complete adjacency graph

Verification

Given: The finite-rank Coxeter, groupoid and label conventions in the Example, followed by the specified $D_{4}$ data.

Goal: Verify the lift, the vertex-group identification, and the eight-element list.

1.1F2given

The pair $(e, J)$ satisfies the defining conditions of a label: $W_{J} e = e W_{J}$ , and the identity has minimum length zero in $W_{J}$ . Apply the lifting assertion in [F2] with starting label $(e, J)$ and arrow $x : J \to I$ . Its endpoint is $(e x, I) = (x, I)$ , and the lift is unique. This proves the first assertion.

2.1F2F3step 1.1

If $x \in {B H}_{J}$ , step 1.1 gives the label $(x, J)$ . Conversely, if $(x, J)$ is a label, the unique arrow from $(e, J)$ to $(x, J)$ has label $e^{- 1} x = x$ , and [F2] makes this a Brink–Howlett arrow $J \to J$ . Thus ${B H}_{J} = {x : (x, J) \in C h a m (J)}$ . By [F3] it is a subgroup of $W$ , and restriction of the quotient map identifies it with $N (W, J)$ .

3.1F4step 2.1∎

The eight rows in [F4] are exactly the labels with second component $J = {2}$ . Substituting their first components into the equality in step 2.1 gives the displayed set. The labels in the table are distinct; since their second components agree, their first components are distinct. Hence the subgroup has exactly eight elements, and the quotient has eight elements by the isomorphism in step 2.1.

Source notes

The article citation attributes the example and its figure. The calculation and group identification above use the completed suppliers; the article's assertion is not used as a proof premise.

Depends on

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF pp. 29–30, §7.2.1 after Figure 4; red labels in Figure 3