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Ribbon generators lift wall-crossing morphisms

PDF p. 22: paragraphs around (5.2)–(5.4)

Statement

Let $Γ$ be a finite-rank spherical Coxeter diagram with vertex set $Γ_{0}$ , Coxeter group $W$ , length $ℓ$ , simple generators $s_{i}$ , and simple roots $α_{i}$ . Let $A$ be its Artin–Tits group with standard generators $σ_{i}$ and positive submonoid $A^{+}$ generated by them. Fix $J \subseteq Γ_{0}$ and a chamber label $(x, I)$ , meaning $W_{J} x = x W_{I}$ and $x$ has minimum length in $x W_{I}$ . Choose $a \in Γ_{0} ∖ I$ , put $T = I \cup {a}$ , and put $K = T ∖ {ι_{T} (a)}$ . Here $W_{L}$ is the subgroup generated by ${s_{i} : i \in L}$ , $w_{L}$ is its longest element, and $ι_{L}$ is its opposition permutation, defined by $w_{L} α_{i} = - α_{ι_{L} (i)}$ .

Let $v_{a, I}$ be the simple wall-crossing element and $μ_{a, I}$ the positive elementary ribbon element. Then $μ_{a, I}$ is the positive reduced-word lift of $v_{a, I}$ and belongs to $A^{+}$ . Moreover $K$ is an associate of $J$ , meaning $W_{K}$ is conjugate to $W_{J}$ , and $μ_{a, I}$ is a morphism $I \to K$ in the ribbon groupoid $R^{'} (J)$ : explicitly, $A_{I}^{+} μ_{a, I} = μ_{a, I} A_{K}^{+}$ , where $A_{L}^{+}$ is the submonoid of $A$ generated by ${σ_{i} : i \in L}$ .

Statement references

Coxeter diagrams: edges, labels, components and finite type

Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups

Artin monoid and Artin group presentations, and the canonical monoid-to-group map

The canonical reflection homomorphism, roots, reflections, and the positive cone

Facts & Assumptions

[F1]

Every standard parabolic is finite. Its longest element is an involution and sends each simple root to the negative of a simple root according to its opposition permutation. Finite parabolic longest elements and disconnected factors Longest parabolic elements and opposition permutation

[F2]

With $T = I \cup {a}$ , the wall element is $v_{a, I} = w_{I} w_{T}$ , and the target in the simple crossing is $T ∖ {ι_{T} (a)}$ . Simplified wall crossing for a finite parabolic Simple wall-crossing elements and maps

[F3]

The ribbon element is $μ_{a, I} = Δ_{I}^{- 1} Δ_{T}$ , where $Δ_{L}$ is the reduced-word lift of $w_{L}$ . Reduced-word lifts are independent of the chosen reduced word and multiply when Coxeter lengths add. Positive elementary ribbon elements Reduced-word lifts and the prefix-order convention

[F4]

A representative $d$ with $ℓ (s_{i} d) > ℓ (d)$ for all $i \in I$ satisfies $ℓ (u d) = ℓ (u) + ℓ (d)$ for every $u \in W_{I}$ . Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification

[F5]

For a simple generator $s_{i}$ , the root-length criterion detects whether multiplication by $s_{i}$ increases length. The reflection representation is faithful; its invariant form identifies conjugates of simple reflections with the reflections in the corresponding image roots. The root-length criterion and faithfulness of the canonical reflection representation Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order conjugation of reflections, part (4)

[F6]

The ribbon groupoid has associates of $J$ as objects and arrows $I \to K$ precisely the labels $β \in A$ satisfying $A_{I}^{+} β = β A_{K}^{+}$ . Ribbon groupoid in spherical type

Proof

Given: The spherical Coxeter and Artin–Tits data, chamber label, and vertex $a$ in the Statement.

Proof technique: direct.

Goal: Identify the positive lift and establish the stated ribbon-groupoid membership.

1.1F1F2F3F4F5

Put $v = w_{I} w_{T}$ . For $i \in I$ , [F1] gives $v^{- 1} α_{i} = w_{T} w_{I} α_{i} = α_{ι_{T} (ι_{I} (i))}$ . The map $i \mapsto ι_{T} (ι_{I} (i))$ is a bijection from $I$ to $K$ : $ι_{I}$ permutes $I$ , and $ι_{T} (I) = T ∖ {ι_{T} (a)}$ . Thus $v^{- 1} α_{i}$ is positive for every $i \in I$ . By the root-length criterion, $ℓ (s_{i} v) > ℓ (v)$ for every such $i$ . Applying [F4] with $u = w_{I}$ gives $ℓ (w_{I} v) = ℓ (w_{I}) + ℓ (v)$ . Since $w_{I}^{2} = 1$ and $w_{I} v = w_{T}$ , the lift multiplication rule gives $Δ_{T} = Δ_{I} σ_{v}$ , where $σ_{v}$ is the reduced-word lift of $v$ . Consequently $μ_{a, I} = Δ_{I}^{- 1} Δ_{T} = σ_{v}$ . A reduced expression for $v$ is a word in the simple generators, so its Artin lift is a product of standard positive generators and belongs to $A^{+}$ . By [F2], $v = v_{a, I}$ .

2.1F3F5step 1.1

For each $i \in I$ , let $k = ι_{T} (ι_{I} (i)) \in K$ . Step 1.1 gives $v α_{k} = α_{i}$ , and these pairs exhaust both $I$ and $K$ . By the reflection conjugation formula and faithfulness, $v s_{k} v^{- 1} = s_{i}$ . The root-length criterion gives $ℓ (v s_{k}) = ℓ (v) + 1$ and, because $v^{- 1} α_{i} = α_{k}$ is positive, $ℓ (s_{i} v) = ℓ (v) + 1$ . Hence the length-additive lift rule yields $σ_{i} σ_{v} = σ_{s_{i} v} = σ_{v s_{k}} = σ_{v} σ_{k}$ . Taking finite products and using the bijection of the two generating sets gives $A_{I}^{+} σ_{v} = σ_{v} A_{K}^{+}$ .

3.1F6step 1.1step 2.1∎

The equality $v s_{k} v^{- 1} = s_{i}$ for the paired generators also gives $v W_{K} v^{- 1} = W_{I}$ . Since $(x, I)$ is a chamber label, $W_{I} = x^{- 1} W_{J} x$ ; therefore $W_{K}$ is conjugate to $W_{J}$ . Thus both $I$ and $K$ are objects of $R^{'} (J)$ . Substituting $μ_{a, I} = σ_{v}$ into the monoid equality from step 2.1 proves that it labels a morphism $I \to K$ , as required.

Source notes

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 22, paragraphs around (5.2)–(5.4)