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Divisibility of positive lifts and weak order

PDF p. 22: Lemma 5.9, opening proof paragraphs; PDF p. 23: Lemma 5.9, opening proof paragraphs

Statement

Let $(W, S)$ be a finite-rank Coxeter system of spherical type, meaning that $W$ is finite. Let $A$ be its Artin–Tits group, $A^{+}$ its submonoid generated by the standard generators $σ_{s}$ , and $π : A \to W$ the homomorphism sending $σ_{s}$ to $s$ . Write $ℓ (w)$ for the minimum length of a word in $S$ representing $w$ . For a reduced expression $w = s_{1} \dots s_{r}$ , define its positive lift by $σ_{w} = σ_{s_{1}} \dots σ_{s_{r}}$ , and put $\tilde{W} = {σ_{w} : w \in W} \subseteq A^{+}$ .

For $a, b \in A^{+}$ , say that $a$ left-divides $b$ if $b = a c$ for some $c \in A^{+}$ . On $W$ use the prefix weak order: $u \leq v$ if $v = u x$ for some $x \in W$ with $ℓ (v) = ℓ (u) + ℓ (x)$ . Restriction of $π$ is an isomorphism from $\tilde{W}$ with left divisibility to $W$ with this order, with inverse $w \mapsto σ_{w}$ . The article calls the latter left weak Bruhat order; the prefix convention here is also called right weak order.

The left-divisibility least common multiple of any two elements of $\tilde{W}$ exists in $A^{+}$ and belongs to $\tilde{W}$ .

Statement references

Artin–Tits group, pure subgroup and positive lifts

Positive Artin and parabolic monoids

Reduced-word lifts and the prefix-order convention

Independent spherical Artin Garside input

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

Order embedding and order isomorphism

Facts & Assumptions

[F1]

Positive lifts are independent of reduced expression, satisfy $π (σ_{w}) = w$ , and obey $σ_{u x} = σ_{u} σ_{x}$ whenever $ℓ (u x) = ℓ (u) + ℓ (x)$ . The stated prefix relation is right weak order. (Reduced-word lifts and the prefix-order convention)

[F2]

In spherical type, the presented positive Artin monoid $M$ embeds as $A^{+}$ ; its Garside element $Δ$ has left divisors precisely the reduced positive lifts, and every pair has a left-divisibility lcm. (Independent spherical Artin Garside input)

[F3]

Word length in the positive generators gives an additive degree on $M$ . Under the embedding this degree transports to $A^{+}$ , and $\deg (σ_{w}) = ℓ (w)$ . (Positive degree and atoms of the presented Artin monoid, Reduced Coxeter lifts in the presented positive monoid, Identifying the presented monoid with its positive image)

[F4]

Every element of $A^{+}$ is represented by a finite positive word. Coxeter length is the minimum length of a representing word; concatenating representative words gives $ℓ (u x) \leq ℓ (u) + ℓ (x)$ . (Positive Artin and parabolic monoids, Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups)

[F5]

An order isomorphism between posets is a bijection that preserves and reflects the order. (Order embedding and order isomorphism)

Proof

Given: The spherical Coxeter and Artin–Tits data in the statement.

Goal: Prove the poset identification and lcm closure.

1.1F1given

The lift map is a section of $π$ by [F1], hence it is injective and $π$ restricts to its inverse on $\tilde{W}$ . If $u \leq v$ , choose $x$ as in the prefix definition. Then [F1] gives $σ_{v} = σ_{u} σ_{x}$ , and $σ_{x} \in A^{+}$ , so $σ_{u}$ left-divides $σ_{v}$ .

2.1F3F4F5step 1.1given

Conversely, suppose $σ_{v} = σ_{u} c$ with $c \in A^{+}$ . Choose a positive word for $c$ of length $k$ and put $x = π (c)$ . Its image is a word of length $k$ for $x$ , so $ℓ (x) \leq k$ by [F4]. Additivity of degree and [F3] give $ℓ (v) = ℓ (u) + k$ . Applying $π$ gives $v = u x$ , and [F4] now gives $ℓ (v) \leq ℓ (u) + ℓ (x) \leq ℓ (u) + k = ℓ (v)$ . Thus both inequalities are equalities and $u \leq v$ . Together with step 1.1 this proves the poset isomorphism.

3.1F2given∎

Let $a, b \in \tilde{W}$ . By [F2] each left-divides $Δ$ , and their left-divisibility lcm $m$ exists. Since $Δ$ is a common upper bound, the defining least-upper-bound property gives that $m$ left-divides $Δ$ . The left divisors of $Δ$ are exactly $\tilde{W}$ by [F2], so $m \in \tilde{W}$ . This proves the remaining assertion.

Depends on

Used by · 1

Ribbon lcms lift two-generator Brink–Howlett elements

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, opening proof paragraphs