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Adjacent chambers and inverse wall-crossing elements

PDF p. 11: Proposition 3.17 and following sentence

Statement

Work in the finite-rank canonical real Coxeter representation and its dual relative chamber arrangement, with arbitrary Coxeter type and arbitrary $J \subseteq Γ_{0}$ . Let $(x, I), (y, K) \in C h a m (J)$ label adjacent chambers $x C_{I}, y C_{K} \subseteq Θ_{J}$ with common closed wall $F$ . There are unique $a \in I^{c}$ , $a^{'} \in K^{c}$ describing this wall, that is, $F = \overline{x C_{I + a}} = \overline{y C_{K + a^{'}}}$ , where closures are taken in $Θ_{J}$ . These indices satisfy $I + a = K + a^{'}$ , the parabolic subgroups $W_{[I, a]} = W_{[K, a^{'}]}$ are finite, and $ω_{a, I} (x, I) = (y, K)$ , $ω_{a^{'}, K} (y, K) = (x, I)$ . In particular $v_{a, I} v_{a^{'}, K} = v_{a^{'}, K} v_{a, I} = e$ . Here $I^{c} = Γ_{0} ∖ I$ and $I + a = I \cup {a}$ . Neither $W_{I}$ nor $W_{I + a}$ is assumed finite.

Facts & Assumptions

[F1]

The unique indices of the common wall, its equal enlarged subset, and its opposite chamber sides are supplied by Common relative facets have one enlarged standard subset. Adjacency uses Chambers, walls and adjacency.

[F2]

The crossing component is finite by Adjacent relative chambers force a finite crossing component.

[F3]

A finite component produces the unique opposite labelled chamber by A finite connected component gives the opposite relative chamber.

[F4]

The reverse component and inverse product are supplied by Componentwise crossings are inverse elements. These products are precisely the elements and maps defined in Simple wall-crossing elements and maps.

[F5]

The coordinate subspaces $Θ_{M}$ and faces $C_{M}$ are defined by Parabolic coordinate subspaces and fundamental faces. The finite-rank geometric convention and the identification of each label $(z, M)$ with the chamber $z C_{M}$ , with unique labels, are supplied by Labelling theorem for relative chambers.

Proof

Given: The finite-rank geometric conventions and chamber identification in [F5], with adjacent relative chambers as in the Statement, in arbitrary Coxeter type and with arbitrary $J$ . Neither $W_{I}$ nor $W_{I + a}$ is assumed finite. Goal: establish the full componentwise formula and its inverse.

Proof technique: direct.

1.1F1F2

Apply [F1] to the prescribed common wall $F$ . It supplies unique wall indices $a \notin I$ , $a^{'} \notin K$ , an equal enlarged subset $L = I + a = K + a^{'}$ , and the displayed wall equality. Apply [F2] to obtain finiteness of $W_{D}$ for $D = [I, a]$ . This is a converse obtained from adjacency, rather than an assumption that the enlarged parabolic is finite.

2.1F1F3F4F5step 1.1

Put $b = ι_{D} (a)$ and $K_{0} = L ∖ {b}$ . By [F3], the chamber across the prescribed wall has label $(x v_{a, I}, K_{0})$ and is therefore the specified adjacent chamber $y C_{K}$ . By label uniqueness in [F5], $y = x v_{a, I}$ and $K = K_{0}$ . Its missing wall index is $b$ ; the uniqueness of the wall index from [F1] gives $a^{'} = b$ . This proves $ω_{a, I} (x, I) = (y, K)$ with the definition in [F4].

3.1F2F4step 1.1step 2.1∎

Apply [F4] with $b = a^{'}$ . It gives $[K, a^{'}] = D$ , $ι_{D} (a^{'}) = a$ and $v_{a^{'}, K} = v_{a, I}^{- 1}$ . Thus $W_{[K, a^{'}]} = W_{D} = W_{[I, a]}$ is finite, both displayed products equal $e$ , and $ω_{a^{'}, K} (y, K) = (x, I)$ by substituting $y = x v_{a, I}$ and $K + a^{'} - a = I$ . These are all the stated assertions.

Source notes

The original article is retained as attribution only. The local argument composes the common-facet, finite-crossing-roots, component-crossing-label and inverse-crossing suppliers. Uniqueness concerns indices describing the specified common wall; enlarged-subset equality alone is not asserted to characterize them. No extension of a theorem requiring a finite enlarged parabolic is used.

Depends on

Used by · 10

Dependency tree · 2 levels

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · Frozen original PDF p. 11, Proposition 3.17 and following sentence