LemmaStatement: AI-generatedProof: AI-generated

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The atomic-quadratic path quotient is faithful

Statement

Let $(W, S)$ be an arbitrary-rank Coxeter system with diagram vertex set $Γ_{0}$ , and fix $J \subseteq Γ_{0}$ . Let $Q$ have the associates of $J$ as vertices and one elementary edge $t_{a, I} : I \to K$ labelled by $v_{a, I} = w_{D ∖ {a}} w_{D}$ for each $a \in Γ_{0} ∖ I$ with $D = [I, a]$ and $W_{D}$ finite, where $b = ι_{D} (a)$ and $K = (I \cup {a}) ∖ {b}$ . Impose all atomic braid relations on these edges and, for every such edge, both ordered quadratic relations $t_{a, I} t_{b, K} = 1_{I}$ and $t_{b, K} t_{a, I} = 1_{K}$ , where $1_{I}$ and $1_{K}$ denote the empty paths. The reverse edge $t_{b, K} : K \to I$ exists under the same elementary-edge rule. Evaluation of labels induces a faithful functor from this path-category quotient to $B H (J)$ , and the quotient is a groupoid.

Facts & Assumptions

[F1]

Use the arbitrary-rank Coxeter system, canonical reflection formulas and simple-root basis of Coxeter-system, roots, length and weak-order conventions, and the associates, elementary morphisms, standard expressions and left-to-right groupoid conventions of Standard expressions and their lengths. All paths are finite.

[F2]

The path category, endpoint-preserving congruence and quotient are defined in Path presentations, congruences and formal inverses. Atomic braid relations equate composable standard expressions of the same morphism by Atomic braid relations; on paths each elementary label is read as its corresponding edge.

[F3]

For a finite set of simple generators in an arbitrary-rank Coxeter system, the group with the restricted Coxeter presentation embeds as the corresponding standard parabolic and its intrinsic length equals ambient length, by Finite-generator standard parabolics and the Bruhat subword property at arbitrary rank, part (1).

[F4]

In a finite-rank Coxeter system, Componentwise crossings are inverse elements applies whenever the crossing component $D$ is finite type: with $A = D ∖ {a}$ , $b = ι_{D} (a)$ and $v = w_{A} w_{D}$ , it gives $ι_{D} (b) = a$ and $w_{D ∖ {b}} w_{D} = v^{- 1}$ . We apply this result inside the finite system on $D$ .

[F5]

Every finite elementary-generator path is congruent to a standard expression of its product modulo atomic braids and both ordered elementary inverse-pair quadratics, by Every finite generator path reduces to a standard expression. Any two standard expressions of the same morphism are connected by atomic braid moves preserving endpoints, by Standard expressions are connected by atomic braid moves.

Proof

Given: The arbitrary-rank Coxeter and groupoid data of [F1], with the path relations specified in the Statement and the inputs [F2], [F3], [F4] and [F5].

Goal: Establish the reverse-edge, groupoid and faithfulness assertions in the Statement.

Proof technique: direct.

1.1F1F3

Fix an elementary edge $t_{a, I}$ and put $D = [I, a]$ , $A = D ∖ {a}$ , $b = ι_{D} (a)$ and $K = (I \cup {a}) ∖ {b}$ . The distinct simple generators indexed by $D$ belong to the finite group $W_{D}$ , so $D$ is finite. By [F3], the Coxeter system with restricted matrix on $D$ identifies with $W_{D}$ and has the ambient length on that subgroup. Its longest elements therefore agree with the ambiently defined $w_{D}$ and $w_{A}$ . By the reflection formulas of [F1], its canonical representation identifies with the restriction to the span of the roots indexed by $D$ ; hence its opposition permutation is the same $ι_{D}$ . Thus the finite-rank hypothesis in [F4] is satisfied inside this subsystem, without a rank restriction on $W$ or a finiteness assumption on $W_{I}$ .

2.1F1F4step 1.1

Apply [F4] in this subsystem, with retained subset $A$ and added vertex $a$ . Its crossing component is $D$ , since $D$ is connected and $A \cup {a} = D$ . It gives $ι_{D} (b) = a$ and $w_{D ∖ {b}} w_{D} = (w_{A} w_{D})^{- 1}$ . In the full diagram, $b \notin K$ and $K \cup {b} = I \cup {a}$ ; the component containing $b \in D$ is still $D$ . Thus $[K, b] = D$ is finite type, and [F1] supplies the elementary edge $t_{b, K}$ with label $v_{b, K} = w_{D ∖ {b}} w_{D} = v_{a, I}^{- 1}$ and target $(K \cup {b}) ∖ {ι_{D} (b)} = I$ . Consequently both ordered quadratic relations in the Statement evaluate to identities.

3.1F1F2step 2.1

Form the endpoint-preserving path congruence and quotient of [F2] using exactly the relations in the Statement. Evaluation sends each edge to its elementary morphism by [F1]. Each atomic relation evaluates equally on its two sides by [F2], and each quadratic relation evaluates to an identity by step 2.1. Thus evaluation descends to a functor on the quotient. Step 2.1 also gives every elementary edge class its two-sided inverse. Reversing the order of a finite path and replacing each edge by its reverse therefore gives a two-sided inverse for its class; the quotient is a groupoid.

4.1F1F5step 2.1step 3.1∎

Consider two paths in the same hom-set whose evaluated labels agree. The relations just specified are the atomic braids and elementary inverse-pair quadratics used in [F5], since step 2.1 identifies every reverse elementary edge and both orders occur in our relation family. By word normalisation in [F5], each path is congruent to a standard expression of its common label. By braid connectivity in [F5], these standard expressions are related by finitely many atomic braid moves with the same endpoints. Hence the original paths are congruent. Evaluation is injective on each hom-set, which is faithfulness. This includes identity labels and empty paths.

Depends on

Used by · 1

Brink–Howlett presentation by atomic and quadratic relations

Dependency tree · 2 levels

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