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Bounded weak-order joins and finite parabolic criteria

Statement

Let $(S, m)$ be a Coxeter matrix on a finite set, $W$ its presented Coxeter group, and $ℓ (w)$ the minimum length of a word in $S$ representing $w$ . The prefix order on $W$ is $u ⪯ v$ exactly when $v = u x$ for some $x \in W$ with $ℓ (v) = ℓ (u) + ℓ (x)$ . An upper bound of $A \subseteq W$ is an element above every member of $A$ ; a join is an upper bound lying below every upper bound. Dually, a meet is a lower bound lying above every lower bound.

Every nonempty subset of $W$ has a meet, and a subset has a join exactly when it has an upper bound. If $W$ is finite, this order is a lattice, meaning that every pair has a meet and join, with top element $w_{S}$ , the unique element of maximum length in $W$ .

For $I \subseteq S$ , put $W_{I} = ⟨ s : s \in I ⟩$ . The set $I$ has a join exactly when $W_{I}$ is finite; then its join is $w_{I}$ , the unique element of maximum length in $W_{I}$ . This includes $I = \emptyset$ , for which $w_{I} = 1$ . Thus a join in an infinite system is asserted only when its set is bounded above.

Statement references

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

The right and left weak orders, intervals, covers, and meets and joins of subsets

Weak order is a meet-semilattice, finite Coxeter groups are lattices, and joins of simple reflections exist exactly for finite parabolics

Facts & Assumptions

[F1]

Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups defines $W$ , its length, and $W_{I} = ⟨ s : s \in I ⟩$ .

[F2]

The right and left weak orders, intervals, covers, and meets and joins of subsets defines right weak order by length-additive factorizations $v = u x$ , and defines bounds, meets, and joins in that order.

[F3]

Weak order is a meet-semilattice, finite Coxeter groups are lattices, and joins of simple reflections exist exactly for finite parabolics (1) says that every nonempty subset has a meet and has a join exactly when it is bounded above in right weak order. Its part (2) gives a lattice with maximum $w_{S}$ when $W$ is finite.

[F4]

Weak order is a meet-semilattice, finite Coxeter groups are lattices, and joins of simple reflections exist exactly for finite parabolics (3) says that $I$ has an upper bound, equivalently a join, precisely when $W_{I}$ is finite; the join is its longest element $w_{I}$ . For $I = \emptyset$ , the join is $1$ .

Proof

Given: A Coxeter matrix $(S, m)$ on a finite set, its presented group $W$ with length $ℓ$ , a subset $A \subseteq W$ , and a subset $I \subseteq S$ .

Goal: Establish the stated meet, join, finite-group, and parabolic assertions for the prefix order.

1.1F2F3F4given

The displayed prefix relation is the right weak order by [F2]. For nonempty $A$ , [F3] therefore supplies its meet and the equivalence between existence of its join and existence of an upper bound. If $A = \emptyset$ , every element is an upper bound, and [F4] supplies the join $1$ of the empty set. Hence the join equivalence holds for every $A \subseteq W$ .

2.1F1F3F4step 1.1given∎

If $W$ is finite, [F3] gives the lattice and maximum $w_{S}$ . For any $I \subseteq S$ , [F1] identifies its standard parabolic subgroup, and [F4] gives the stated finiteness criterion and join value, including the empty case. The final boundedness qualification follows from step 1.1.

Depends on

Used by · 1

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

Dependency tree · 2 levels

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