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Wall-crossing elements are Brink–Howlett morphisms

PDF p. 15: Lemma 4.2

Statement

Let $(W, S)$ be a Coxeter system with simple generators $s_{i}$ and simple roots $α_{i}$ indexed by the vertices $Γ_{0}$ of its Coxeter diagram. Fix $J \subseteq Γ_{0}$ , and let $I$ be an associate of $J$ , meaning that $W_{I} = ⟨ s_{i} : i \in I ⟩$ is conjugate to $W_{J}$ in $W$ . Choose $a \in Γ_{0} ∖ I$ , and let $D = [I, a]$ be the component containing $a$ in the diagram induced on $I \cup {a}$ . Suppose $W_{D}$ is finite. For a finite standard parabolic $W_{L}$ , write $w_{L}$ for its longest element and $ι_{L}$ for the permutation of $L$ such that $w_{L} α_{i} = - α_{ι_{L} (i)}$ . Put $E = D ∖ {a}$ , $K = (I \cup {a}) ∖ {ι_{D} (a)}$ and $v_{a, I} = w_{E} w_{D}$ .

Then $K$ is an associate of $J$ , and $v_{a, I}$ is a Brink–Howlett morphism from $I$ to $K$ : explicitly, ${v_{a, I} α_{k} : k \in K} = {α_{i} : i \in I}$ .

Statement references

Brink–Howlett groupoid

Connected component notation [I,a]

Longest parabolic elements and opposition permutation

Associates of a parabolic subset

Coxeter-system, roots, length and weak-order conventions

Facts & Assumptions

[F1]

For finite $W_{L}$ , the longest element and opposition permutation are defined by $w_{L} α_{i} = - α_{ι_{L} (i)}$ . Longest parabolic elements and opposition permutation

[F2]

A standard parabolic on a finite subset of the simple generators is the Coxeter group for its restricted matrix, with its intrinsic length equal to ambient length. Finite-generator standard parabolics and the Bruhat subword property at arbitrary rank

[F3]

In a finite Coxeter system, its longest element is an involution and conjugates each simple generator to the generator indexed by its opposition image. The longest element as the opposition of the chamber, and longest elements of finite parabolics

[F4]

Distinct diagram vertices with no edge have Coxeter exponent two. Their generators commute, and the reflection formula makes either generator fix the other's simple root. Coxeter-system, roots, length and weak-order conventions

[F5]

The component $D$ contains $a$ and has no diagram edge to any vertex of $(I \cup {a}) ∖ D$ . Connected component notation [I,a]

[F6]

An associate is a subset with conjugate standard parabolic subgroup; a Brink–Howlett morphism $I \to K$ is an element sending the simple-root set indexed by $K$ onto that indexed by $I$ . Associates of a parabolic subset Brink–Howlett groupoid

Proof

Given: The Coxeter system, subsets and elements in the Statement.

Proof technique: direct.

Goal: Show that $v_{a, I}$ maps the simple roots and generators indexed by $K$ bijectively to those indexed by $I$ .

1.1F1F2F3given

The distinct simple generators $s_{i}$ indexed by $D$ belong to the finite group $W_{D}$ , so $D$ is finite. Hence $E \subseteq D$ is finite, and $W_{E} \leq W_{D}$ is finite. By [F2], apply [F3] to the finite Coxeter systems on $D$ and $E$ . Their longest elements are involutions, and their opposition permutations are involutions: applying $w_{L}^{2} = 1$ to $α_{i}$ and using [F1] gives $α_{i} = α_{ι_{L}^{2} (i)}$ , so $ι_{L}^{2} (i) = i$ . The same application of [F3] gives $w_{L} s_{i} w_{L}^{- 1} = s_{ι_{L} (i)}$ for $i \in L$ , with $L = D$ or $E$ .

2.1F1F5step 1.1given

Put $R = I ∖ D$ . Since $a \in D$ and $D \subseteq I \cup {a}$ , one has $I = R ⊔ E$ and $K = R ⊔ (D ∖ {ι_{D} (a)})$ . The involution $ι_{D}$ maps $D ∖ {ι_{D} (a)}$ bijectively onto $E$ . For $k$ in this set, [F1] therefore gives $v_{a, I} α_{k} = α_{ι_{E} (ι_{D} (k))}$ , and step 1.1 gives $v_{a, I} s_{k} v_{a, I}^{- 1} = s_{ι_{E} (ι_{D} (k))}$ . These images run bijectively over the roots and generators indexed by $E$ .

3.1F4F5step 2.1

For $k \in R$ , there is no edge from $k$ to a vertex of $D$ by [F5]. Hence every generator in $D$ fixes $α_{k}$ and commutes with $s_{k}$ by [F4]. Since $w_{E}, w_{D} \in W_{D}$ , it follows that $v_{a, I} α_{k} = α_{k}$ and $v_{a, I} s_{k} v_{a, I}^{- 1} = s_{k}$ . Combining this with step 2.1 yields both $v_{a, I} {α_{k} : k \in K} = {α_{i} : i \in I}$ and $v_{a, I} W_{K} v_{a, I}^{- 1} = W_{I}$ .

4.1F6step 3.1given∎

Because $W_{I}$ is conjugate to $W_{J}$ , step 3.1 shows that $W_{K}$ is also conjugate to $W_{J}$ . Thus $K$ is an associate of $J$ , and the root-set equality of step 3.1 makes $v_{a, I}$ a morphism $I \to K$ by [F6]. This proves the claim.

Depends on

Used by · 1

Standard expressions and their lengths

Dependency tree · 2 levels

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 15, Lemma 4.2