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Four colour orbits of eight D4 chambers

PDF p. 28: Figure 2; §7.2.1; PDF p. 29: Figure 3; §7.2.1, penultimate paragraph

Example

Take the $D_{4}$ Coxeter diagram with central vertex $2$ and $J = {2}$ . Let $W_{I}$ denote the subgroup generated by the simple reflections indexed by $I$ , and put $N (W, J) = {Norm}_{W} (W_{J}) / W_{J}$ . The relative chambers are labelled by pairs $(x, I)$ with $W_{J} x = x W_{I}$ and $x$ the minimum-length element of $x W_{I}$ ; write $(x, i)$ for $(x, {i})$ .

The action of $N (W, J)$ on these chambers has four orbits, each containing eight chambers. Their second labels are respectively ${1}, {2}, {3}, {4}$ : these are precisely the associates of $J$ , meaning the objects of the Brink–Howlett groupoid. In Figures 2–3 of the article, equal colours denote equal second labels: black denotes $1$ , red $2$ , green $3$ and blue $4$ .

The universal Brink–Howlett covering sends the object $(x, I)$ to $I$ and the arrow from $(x, I)$ to $(y, K)$ labelled $x^{- 1} y$ to the arrow with the same label $x^{- 1} y$ from $I$ to $K$ .

Example references

D4 chamber labels and complete adjacency graph

Normaliser action on labels, orbits and adjacency

Universal Brink–Howlett covering and its deck group

Facts & Assumptions

[F1]

The completed chamber table lists all 32 relative chambers, numbered 1 through 32, with their labels $(x, i)$ . D4 chamber labels and complete adjacency graph

[F2]

The normaliser quotient acts on chamber labels, preserves their second component, and is transitive on labels with any fixed second component. Normaliser action on labels, orbits and adjacency

[F3]

The universal Brink–Howlett groupoid has the chamber labels as objects and one arrow from $(x, I)$ to $(y, K)$ labelled $x^{- 1} y$ . The base objects are the associates of $J$ , and the covering sends $(x, I)$ to $I$ and retains the arrow label. Universal Brink–Howlett covering and its deck group

Verification

Given: The $D_{4}$ data and chamber labels in the Example.

Goal: Verify the four orbits, their sizes, and the stated covering map.

1.1F1F2

Read the second components in the complete table [F1]. For $i = 1, 2, 3, 4$ , respectively, the numbered chambers are ${2, 3, 4, 6, 27, 29, 30, 31}$ , ${1, 5, 7, 9, 24, 26, 28, 32}$ , ${10, 12, 13, 16, 18, 19, 22, 25}$ , and ${8, 11, 14, 15, 17, 20, 21, 23}$ . These four disjoint sets exhaust the table and each has eight entries. By [F2], each set is one orbit and different sets are different orbits.

2.1F3step 1.1∎

By [F3], the covering maps each chamber label to its second component and preserves the stated arrow label; its base objects are exactly the associates of $J$ . The four nonempty second-component sets in step 1.1 therefore give exactly the four associates listed in the Example. This verifies the orbit count, orbit sizes and covering formula.

Source notes

The article's paragraph states that the two figures use the same colour for chambers exactly when their second labels agree. The displayed key is read from Figure 3. The orbit calculation and covering assertion above use the completed suppliers, not the article's assertion as a proof premise.

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Used by · 2

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF pp. 28–29, Figures 2–3 and §7.2.1, penultimate paragraph on p. 29