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Ribbon lcms lift two-generator Brink–Howlett elements

PDF p. 22: Lemma 5.9; PDF p. 23: Lemma 5.9

Statement

Let $(W, S)$ be a finite-rank spherical Coxeter system, so $W$ is finite, and let $A$ be its Artin group with positive submonoid $A^{+}$ and natural homomorphism $π : A \to W$ . Fix $J \subseteq S$ and an associate $I$ of $J$ , meaning that $W_{I}$ is conjugate to $W_{J}$ . For $a \in S ∖ I$ , let $μ_{a, I}$ be the elementary ribbon and $v_{a, I}$ its corresponding elementary Brink–Howlett morphism, with the conventions of Reduced ribbon groupoid and Standard expressions and their lengths. A positive lift $σ_{w}$ of $w \in W$ is the Artin word obtained from a reduced Coxeter word for $w$ .

For distinct $a, b \in S ∖ I$ , let $q = v_{a, b, I}$ be the two-generator morphism of Two-generator minimal common left multiple, and let $L = μ_{a, I} \lor μ_{b, I}$ be the least common multiple for left divisibility in $A^{+}$ . Its two elementary-ribbon expressions, beginning respectively with $μ_{a, I}$ and $μ_{b, I}$ , are precisely the factorwise positive lifts of the two standard expressions of $q$ . Let $m$ and $n$ be the respective numbers of elementary factors in these two ribbon expressions. Then $m = n$ , and $π (L) = q$ . Moreover $q$ is the join of $v_{a, I}$ and $v_{b, I}$ in the prefix weak order: $u \leq v$ means $v = u x$ with $ℓ (v) = ℓ (u) + ℓ (x)$ . This is the order called weak left Bruhat order in the article.

For $a = b$ , the lcm is $μ_{a, I}$ and the join is $v_{a, I}$ ; the assertion of two distinct expressions is not made in this case.

Facts & Assumptions

[F1]

Every elementary ribbon $μ_{c, H}$ is the positive lift of the corresponding elementary Brink–Howlett morphism $v_{c, H}$ , with the same source and target. Ribbon generators lift wall-crossing morphisms Reduced ribbon groupoid Standard expressions and their lengths

[F2]

For distinct $a, b$ , the ribbon lcm $L$ has exactly two elementary-arrow paths from $I$ , beginning respectively with the two specified ribbons. Two ribbon-lcm expressions and presentation

[F3]

For $T \subseteq S$ , the parabolic roots are $Φ_{T} = {u α_{s} : u \in W_{T}, s \in T}$ , and the ambient roots are $Φ = {u α_{s} : u \in W, s \in S}$ . Coxeter-system, roots, length and weak-order conventions Standard parabolic subgroups and parabolic roots Under the hypothesis $∣ Φ_{I \cup {a, b}} ∖ Φ_{I} ∣ < \infty$ , Two-generator minimal common left multiple supplies $q$ and its two standard expressions, one starting with each elementary morphism, and says $q$ is a prefix of every Brink–Howlett morphism from $I$ having both specified prefixes. These expressions have the same number of factors.

[F4]

Positive lifts form a poset isomorphic under $π$ to $W$ with prefix weak order; the lcm in $A^{+}$ of two positive lifts is again a positive lift. Divisibility of positive lifts and weak order

[F5]

Positive lifts multiply over length-additive Coxeter products. The positive-word degree is additive and the degree of $σ_{w}$ is $ℓ (w)$ . Reduced-word lifts and the prefix-order convention Positive degree and atoms of the presented Artin monoid Divisibility of positive lifts and weak order

Proof

Given: The spherical system, the associate $I$ , and vertices $a, b$ of the Statement.

Proof technique: direct.

Goal: Identify the two paths, the lcm image, and the weak-order join, including the diagonal case.

1.1F1F2F3F4F5given

Suppose $a \neq b$ . The root set $Φ$ in [F3] is the image of the finite set $W \times S$ , so it is finite. Since $W_{I \cup {a, b}} \subseteq W$ , the set $Φ_{I \cup {a, b}} ∖ Φ_{I}$ is a subset of $Φ$ and is finite. Thus the hypothesis of the two-generator lemma in [F3] holds. By [F1] and [F4], $L$ is a positive lift, say $L = σ_{w}$ . Choose either elementary-ribbon path for $L$ from [F2] and replace each factor by its image elementary morphism. The source and target match by [F1], so their product is a Brink–Howlett morphism $w$ from $I$ . If the factors are $v_{1}, \dots, v_{r}$ , additivity of positive degree gives $ℓ (w) = \sum_{i} ℓ (v_{i})$ by [F5]. Hence this is a standard expression by the definition in Standard expressions and their lengths. In particular $w$ has both $v_{a, I}$ and $v_{b, I}$ as prefixes, using the two paths and their length additivity. By [F3], $q$ is a prefix of $w$ .

2.1F1F3F4F5step 1.1

Each standard expression of $q$ lifts factorwise to $σ_{q}$ by repeated length-additive multiplication. The expressions start with the two elementary morphisms, so [F1] makes both $μ_{a, I}$ and $μ_{b, I}$ left divisors of $σ_{q}$ . The lcm property yields $L$ left-dividing $σ_{q}$ , and [F4] translates this to $w \leq q$ . Step 1.1 gives $q \leq w$ ; antisymmetry of the prefix order gives $w = q$ , and therefore $L = σ_{q}$ and $π (L) = q$ .

3.1F1F2F3F4step 2.1

The factorwise lifts of the two standard expressions of $q$ are elementary-ribbon paths for $L$ , starting with different specified arrows. By the exactly-two assertion in [F2], they are precisely the two paths there. Their factor counts are equal by [F3], proving $m = n$ . For any common upper bound $z$ of $v_{a, I}$ and $v_{b, I}$ in prefix order, [F4] makes $σ_{z}$ a common multiple of the two ribbons. Thus $L$ divides $σ_{z}$ , and [F4] gives $q \leq z$ . The two original elements lie below $q$ by [F3], so $q$ is their join.

4.1F4given∎

If $a = b$ , the lcm of an element with itself is that element, and the join of an order element with itself is that element. This gives the diagonal assertions.

Source notes

The article citation records attribution. The proof uses the listed completed suppliers; it does not use the article's Lemma 5.9 as a premise.

Depends on

Used by · 1

Deligne-to-ribbon functor is a covering

Dependency tree · 2 levels

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Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF pp. 22–23, Lemma 5.9