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Simplified wall crossing for a finite parabolic

PDF p. 11: Lemma 3.20(4); (3.21)

Statement

Let $(W, S)$ be a Coxeter system indexed by the vertices $Γ_{0}$ of its Coxeter diagram, let $J \subseteq Γ_{0}$ , and let $(x, I)$ be a chamber label: $W_{J} x = x W_{I}$ and $x$ has minimum length in $x W_{I}$ . Choose $a \in Γ_{0} ∖ I$ and put $K = I \cup {a}$ . Suppose $W_{K}$ is finite. For a finite parabolic $W_{L}$ , write $w_{L}$ for its longest element and $ι_{L}$ for its opposition permutation, characterized by $w_{L} α_{i} = - α_{ι_{L} (i)}$ for $i \in L$ .

With the simple wall-crossing element $v_{a, I}$ and pair $ω_{a, I} (x, I)$ defined in Simple wall-crossing elements and maps, one has $v_{a, I} = w_{I} w_{K}$ and $ω_{a, I} (x, I) = (x w_{I} w_{K}, K ∖ {ι_{K} (a)})$ .

Facts & Assumptions

[F1]

The simple crossing is defined by $v_{a, I} = w_{D ∖ {a}} w_{D}$ and $ω_{a, I} (x, I) = (x v_{a, I}, K ∖ {ι_{D} (a)})$ . Simple wall-crossing elements and maps

[F2]

A finite standard parabolic has a finite indexing set; the restricted Coxeter system has its ambient length, and its geometric representation agrees with the ambient representation on the span of its simple roots. Finite-generator standard parabolics and the Bruhat subword property at arbitrary rank Coxeter-system, roots, length and weak-order conventions

[F3]

For a finite-rank Coxeter system, longest elements factor over diagram components, those factors commute, and common components cancel in the product of longest elements. Finite parabolic longest elements and disconnected factors

[F4]

For a finite parabolic, the longest element sends each simple root to the negative of the simple root indexed by its opposition permutation. Longest parabolic elements and opposition permutation

Proof

Given: The Coxeter system, chamber label, vertex $a$ , and finite parabolic $W_{K}$ in the Statement. Put $D = [I, a]$ , the component containing $a$ of the diagram induced on $K$ , and put $H = K ∖ D$ .

Proof technique: direct.

Goal: Prove the two displayed formulas in the Statement.

1.1F1F2F3

Since $W_{K}$ is finite, [F2] makes $K$ finite and permits application of [F3] to the Coxeter system on $K$ . The set $H$ is a union of components of the diagram induced on $K$ . It is also a union of components of the diagram induced on $I$ : deleting $a$ cannot create an edge from $H$ to $D ∖ {a}$ , and the components contained in $H$ are unchanged. Since $I ∖ H = D ∖ {a}$ and $K ∖ H = D$ , the cancellation formula in [F3] gives $w_{I} w_{K} = w_{D ∖ {a}} w_{D} = v_{a, I}$ . All the displayed parabolics are finite subgroups of $W_{K}$ .

2.1F1F2F3F4step 1.1∎

By the component factorization in [F3], $w_{K} = w_{H} w_{D}$ . There are no diagram edges between $H$ and $D$ , so the defining reflection formula fixes every simple root indexed by $D$ under every generator indexed by $H$ . In particular, using [F4], $w_{K} α_{a} = w_{H} (- α_{ι_{D} (a)}) = - α_{ι_{D} (a)}$ . The simple roots are distinct basis vectors, so comparison with $w_{K} α_{a} = - α_{ι_{K} (a)}$ yields $ι_{K} (a) = ι_{D} (a)$ . Substituting this equality and step 1.1 into [F1] gives the asserted formula for $ω_{a, I} (x, I)$ .

Depends on

Used by · 4

Dependency tree · 2 levels

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 11, Lemma 3.20(4) and equation (3.21)