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Translation between relative arrangements in the geodesic argument

PDF p. 14: Proposition 3.29, final proof paragraph

Statement

Work with the finite-rank real Coxeter representation and relative arrangements, with arbitrary Coxeter type and $J \subseteq Γ_{0}$ . Let $(z, L) \in C h a m (J)$ . Then $z^{- 1}$ maps the relative arrangement for $J$ bijectively to that for $L$ , including their cone interiors and distinct relative hyperplanes. It induces a bijection of chamber labels sending $(z, L)$ to $(e, L)$ : the image of $(x, I)$ is $(x^{'}, I)$ , where $x^{'}$ is the unique minimum-length element of $z^{- 1} x W_{I}$ .

Under this bijection every chamber path $p$ maps to a path of the same length; geodesicity is preserved and reflected. Each simple wall crossing retains its index $a$ and element $v_{a, I}$ , so the wall word element satisfies $v_{z^{- 1} p} = v_{p}$ .

Statement references

Labelling theorem for relative chambers

Algebraic chamber-label set

Tits cone intersections, relative roots and arrangements

Chambers, walls and adjacency

Chamber paths, concatenation and wall words

Geodesic chamber paths

Simple wall-crossing elements and maps

Real Coxeter representation and contragredient notation

Parabolic root subsystems and unique coset minima

Facts & Assumptions

[F1]

The label condition is $W_{J} z = z W_{L}$ ; the chamber labelled $(x, I)$ is $x C_{I}$ , and this labelling is bijective. Each right coset $g W_{I}$ has a unique minimum-length representative. Algebraic chamber-label set Labelling theorem for relative chambers Parabolic root subsystems and unique coset minima

[F2]

The relative space $Θ_{K}$ is the common zero set of the simple roots indexed by $K$ ; the relative cone is $C o n e \cap Θ_{K}$ , and its arrangement consists of distinct proper root-hyperplane traces. The contragredient action sends $H_{α}$ to $H_{w α}$ , and the Tits cone is $W$ -invariant. Tits cone intersections, relative roots and arrangements Real Coxeter representation and contragredient notation Basic Tits-cone properties

[F3]

Chambers are complement components and adjacency is sharing a codimension-one wall. A path is a finite sequence of consecutively adjacent chamber labels; its length counts consecutive pairs. A geodesic minimizes this length among paths with the same endpoints. Chambers, walls and adjacency Chamber paths, concatenation and wall words Geodesic chamber paths

[F4]

For an ordered adjacent pair starting at $(x, I)$ , the adjacent-wall formula gives its unique crossing index $a \notin I$ , its wall $\overline{x C_{I \cup {a}}}$ , and its target $(x v_{a, I}, K)$ . The element $v_{a, I}$ and $K$ depend only on $a$ and $I$ . The wall word is the ordered product of these elements, or $e$ for a one-chamber path. Adjacent chambers and inverse wall-crossing elements Simple wall-crossing elements and maps Chamber paths, concatenation and wall words

Proof

Given: The finite-rank representation, $J$ and $(z, L)$ in the Statement, and a chamber path $p$ in the $J$ -arrangement. Goal: Prove the stated arrangement and label bijections and the path assertions.

Proof technique: direct.

1.1F1F2

For any subset $K$ , the fixed space of $W_{K}$ in $Θ$ is $Θ_{K}$ : the reflection formula in [F2] shows that the simple reflection indexed by $k$ fixes a functional exactly when it vanishes on $α_{k}$ , and the generators then fix precisely their common zero set. From $z^{- 1} W_{J} z = W_{L}$ in [F1], conjugation of fixed spaces gives $z^{- 1} Θ_{J} = Θ_{L}$ . By [F2], $z^{- 1}$ preserves the Tits cone and permutes root hyperplanes. It therefore maps $C o n e (J)$ onto $C o n e (L)$ and, being a linear homeomorphism of the relative spaces, maps their relative interiors onto one another. It also bijects their sets of distinct proper root-hyperplane traces; coincident traces remain coincident under this bijection.

2.1F1F2F3step 1.1

The homeomorphism in step 1.1 maps complement components bijectively, and preserves closures and affine dimensions. Hence it maps walls and adjacency bijectively. For a label $(x, I)$ , let $x^{'}$ be the unique minimum of $z^{- 1} x W_{I}$ and write $z^{- 1} x = x^{'} u$ with $u \in W_{I}$ . Every generator indexed by $I$ fixes $C_{I}$ pointwise, by the reflection formula, so $z^{- 1} x C_{I} = x^{'} C_{I}$ . The label equality $W_{J} x = x W_{I}$ and $z^{- 1} W_{J} z = W_{L}$ imply $W_{L} x^{'} = x^{'} W_{I}$ . Thus $(x^{'}, I)$ is a label for $L$ , and it is the image label by [F1]. For $(x, I) = (z, L)$ the coset is $W_{L}$ , whose minimum is $e$ , so the image is $(e, L)$ .

3.1F2F4step 2.1

Suppose a crossing from $(x, I)$ has index $a$ . With $z^{- 1} x = x^{'} u$ as in step 2.1, every element of $W_{I}$ also fixes $C_{I \cup {a}}$ pointwise. Thus the image of its wall is $\overline{x^{'} C_{I \cup {a}}}$ . The image chambers are adjacent by step 2.1, and the uniqueness in [F4] identifies their crossing index as the same $a$ . Consequently their crossing element is the same $v_{a, I}$ , and their target is $(x^{'} v_{a, I}, K)$ .

4.1F3F4step 2.1step 3.1∎

Applying the chamber bijection to each entry of $p$ produces a path of the same length, and the inverse homeomorphism gives the reverse correspondence for paths between the corresponding endpoints. Therefore a shorter path exists between one pair of endpoints exactly when one exists between the other pair, proving preservation and reflection of geodesicity. By step 3.1 the ordered crossing elements agree term by term, including the empty product for a one-chamber path; hence $v_{z^{- 1} p} = v_{p}$ . This proves all assertions.

Source notes

The article reference attributes the assertion. The argument above uses the declared completed suppliers and does not use the article assertion as a proof premise.

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 14, Proposition 3.29, final proof paragraph