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Meets and joins are not intersection and union of inversion sets: the $A_{2}$ counterexample

Statement refuted

Statement refuted. For every finite Coxeter system $(W, S)$ , every $u, v \in W$ for which $u \land v$ and $u \lor v$ exist satisfy

$N ((u \land v)^{- 1}) = N (u^{- 1}) \cap N (v^{- 1}), N ((u \lor v)^{- 1}) = N (u^{- 1}) \cup N (v^{- 1})$

(with $N$ the inversion sets of The geometric inversion set $N (w)$ of an element of a Coxeter group), i.e. meets and joins are computed by intersection and union of inversion sets.

Counterexample (type $A_{2}$ , $W = S_{3}$ ). Let $S = {s, t}$ , $m (s, t) = 3$ , and let $α_{s}, α_{t}, α_{s} + α_{t}$ be the positive roots of $A_{2}$ , so that $N (w^{- 1}) \in {\emptyset, {α_{s}}, {α_{t}}, {α_{s}, α_{s} + α_{t}}, {α_{t}, α_{s} + α_{t}}, Φ_{+}}$ for $w \in W = {1, s, t, s t, t s, w_{0}}$ as in All meets and joins of the right weak order of $A_{2}$ ( $S_{3}$ ), with the left order and the inversion sets compared. Then:

(i) $s \lor t = w_{0}$ and $N (w_{0}^{- 1}) = Φ_{+} = {α_{s}, α_{t}, α_{s} + α_{t}}$ , while $N (s^{- 1}) \cup N (t^{- 1}) = {α_{s}} \cup {α_{t}} = {α_{s}, α_{t}} ⊊ Φ_{+}$ ; the union is not even the inversion set of an element of $W$ , and in particular the join is strictly larger than the union.

(ii) $s t \land t s = 1$ and $N (1) = \emptyset$ , while $N ((s t)^{- 1}) \cap N ((t s)^{- 1}) = {α_{s}, α_{s} + α_{t}} \cap {α_{t}, α_{s} + α_{t}} = {α_{s} + α_{t}} \neq \emptyset$ ; the intersection is not the inversion set of an element of $W$ , and in particular the meet is strictly smaller than the intersection.

Correct statement. By Weak order is a partial order with finite graded intervals; covers and the inversion-set criterion (4), $w \leq_{R} u, v$ if and only if $N (w^{- 1}) \subseteq N (u^{- 1}) \cap N (v^{- 1})$ ; hence $u \land v$ is the greatest element whose inversion set is contained in the intersection, and dually $u \lor v$ is the least element whose inversion set contains the union. Containment, and not equality, is the correct order-theoretic relation.

Facts & Assumptions

Given: The Coxeter matrix of type $A_{2}$ : $S = {s, t}$ , $m (s, t) = 3$ ; the presented group $W$ with length $ℓ$ , the weak orders $\leq_{R}, \leq_{L}$ as in The right and left weak orders, intervals, covers, and meets and joins of subsets; the reflection representation $ρ$ on $V = R^{S}$ with simple roots $α_{s} = e_{s}$ , $α_{t} = e_{t}$ and root system $Φ = Φ_{+} ⊔ Φ_{-}$ ; and $w_{0} = s t s = t s t$ .

[F1]

The geometric inversion set $N (w)$ of an element of a Coxeter group (1): $N (w) = {α \in Φ_{+} : ρ (w) α \in Φ_{-}}$ .

[F2]

All meets and joins of the right weak order of $A_{2}$ ( $S_{3}$ ), with the left order and the inversion sets compared: $W = {1, s, t, s t, t s, w_{0}}$ with $ℓ (1) = 0$ , $ℓ (s) = ℓ (t) = 1$ , $ℓ (s t) = ℓ (t s) = 2$ , $ℓ (w_{0}) = 3$ ; and $w_{0} = s t s = t s t$ .

[F3]

All meets and joins of the right weak order of $A_{2}$ ( $S_{3}$ ), with the left order and the inversion sets compared: the six inversion sets $N (w^{- 1})$ for $w = 1, s, t, s t, t s, w_{0}$ are $\emptyset$ , ${α_{s}}$ , ${α_{t}}$ , ${α_{s}, α_{s} + α_{t}}$ , ${α_{t}, α_{s} + α_{t}}$ and $Φ_{+} = {α_{s}, α_{t}, α_{s} + α_{t}}$ .

[F4]

All meets and joins of the right weak order of $A_{2}$ ( $S_{3}$ ), with the left order and the inversion sets compared: $s \lor t = w_{0}$ and $s t \land t s = 1$ , with their universal bound properties recorded there.

[F5]

Weak order is a partial order with finite graded intervals; covers and the inversion-set criterion (4): for all $u, v$ , $u \leq_{R} v ⟺ N (u^{- 1}) \subseteq N (v^{- 1})$ .

[F6]

Weak order is a meet-semilattice, finite Coxeter groups are lattices, and joins of simple reflections exist exactly for finite parabolics (2): if $W$ is finite, every subset has a meet and a join.

[F7]

The right and left weak orders, intervals, covers, and meets and joins of subsets (3): a right meet is a greatest lower bound in $\leq_{R}$ .

[F8]

The right and left weak orders, intervals, covers, and meets and joins of subsets (3): a right join is a least upper bound in $\leq_{R}$ .

Counterexample

1.1F1F2F3givenalgebra

The $A_{2}$ data: $W = {1, s, t, s t, t s, w_{0}}$ with the lengths of Fact [F2], and the six inversion sets $N (w^{- 1})$ of Fact [F3]; in particular, no other subset of $Φ_{+}$ displayed below occurs as an inversion set $N (w^{- 1})$ .

2.1F1F4F8step 1.1givenalgebra

Failure of the join equality: $s \lor t = w_{0}$ by [F4] and $N (w_{0}^{- 1}) = Φ_{+} = {α_{s}, α_{t}, α_{s} + α_{t}}$ by step 1.1, while $N (s^{- 1}) \cup N (t^{- 1}) = {α_{s}} \cup {α_{t}} = {α_{s}, α_{t}}$ . The union ${α_{s}, α_{t}}$ is not among the six sets of step 1.1, so it is not the inversion set $N (w^{- 1})$ of any element $w \in W$ ; in particular $N ((s \lor t)^{- 1}) = Φ_{+} \neq {α_{s}, α_{t}} = N (s^{- 1}) \cup N (t^{- 1})$ , refuting the join half of the displayed statement.

2.2F1F4F7step 1.1givenalgebra

Failure of the meet equality: $s t \land t s = 1$ by [F4] and $N (1) = \emptyset$ by step 1.1, while $N ((s t)^{- 1}) \cap N ((t s)^{- 1}) = {α_{s}, α_{s} + α_{t}} \cap {α_{t}, α_{s} + α_{t}} = {α_{s} + α_{t}}$ is nonempty; this intersection is not among the six sets of step 1.1 either, so it is not the inversion set $N (w^{- 1})$ of any element, and $N ((s t \land t s)^{- 1}) = \emptyset \neq {α_{s} + α_{t}} = N ((s t)^{- 1}) \cap N ((t s)^{- 1})$ , refuting the meet half of the displayed statement.

3.1F5F6F7F8step 2.1step 2.2givenalgebra∎

The corrected containment statement: by the criterion [F5], for every $w \in W$ , $w \leq_{R} u$ and $w \leq_{R} v$ if and only if $N (w^{- 1}) \subseteq N (u^{- 1})$ and $N (w^{- 1}) \subseteq N (v^{- 1})$ , equivalently $N (w^{- 1}) \subseteq N (u^{- 1}) \cap N (v^{- 1})$ . Since $W$ is finite, [F6] guarantees that $u \land v$ and $u \lor v$ exist; by [F7], the meet is the greatest such lower bound. Dually, [F5] gives $w \geq_{R} u, v$ if and only if $N (w^{- 1}) \supseteq N (u^{- 1}) \cup N (v^{- 1})$ , and [F8] makes the join the least such upper bound. Thus the meet inversion set is the greatest inversion set contained in the intersection, and the join inversion set is the least inversion set containing the union; steps 2.1 and 2.2 show both containments can be strict. No Choice is used.

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Sources

Anders Bjorner and Francesco Brenti, Combinatorics of Coxeter Groups (Graduate Texts in Mathematics 231, Springer 2005; author-hosted complete PDF) (standard reference, not scraped)
Nathan Reading and David E. Speyer, Cambrian fans (J. Eur. Math. Soc. 11 (2009) 407-447; arXiv:math/0606201v2) (standard reference, not scraped)

Meets and joins are not intersection and union of inversion sets: the A2A2​ counterexample