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A set of two reflections of A2 that fails both closure and the segment criterion

Statement

Let $(W, S)$ be the Coxeter system of type $A_{2}$ , with $S = {s, t}$ and $m (s, t) = 3$ . Its reflections and corresponding positive roots in angular order are $u_{1} = s, u_{2} = s t s, u_{3} = t, β_{u_{1}} = e_{s}, β_{u_{2}} = e_{s} + e_{t}, β_{u_{3}} = e_{t}$ (Plane subsystems, their canonical generators, and the angular order of their roots (2), The real Coxeter form, its radical, reflections, and form-preserving maps (2), The inversion formula $∣ N (w) ∣ = ℓ (w)$ , the root-reflection dictionary and strong exchange (1)). Put $I : = {e_{s}, e_{t}} \subseteq Φ_{+} .$

(i) $I$ is not $N (w)$ for any $w \in W$ ; explicitly, $N (1) = \emptyset, N (s) = {e_{s}}, N (t) = {e_{t}}, N (s t) = {e_{t}, e_{s} + e_{t}}, N (t s) = {e_{s}, e_{s} + e_{t}}, N (s t s) = Φ_{+} .$

(ii) A set is closed under positive rank-two combinations when it contains every root $a α + b β \in Φ_{+}$ with $a, b > 0$ and $α, β$ in the set. Then $I$ is not closed: it contains $e_{s}, e_{t}$ but omits their root $e_{s} + e_{t}$ . Its complement $Φ_{+} ∖ I = {e_{s} + e_{t}}$ is closed.

(iii) $I$ is neither an initial nor a final segment of $(β_{u_{1}}, β_{u_{2}}, β_{u_{3}})$ . Thus the rank-two segment criterion of Finite inversion sets are recognized by their rank-two initial or final segments (1)(ii) rejects $I$ .

(iv) By contrast, $I^{'} : = {e_{s}, e_{s} + e_{t}}$ is the initial segment $(β_{u_{1}}, β_{u_{2}})$ and equals $N (t s)$ .

(v) The complement $I^{c} : = {e_{s} + e_{t}}$ is closed under positive rank-two combinations but is not an inversion set. Thus closure of a set alone is insufficient.

Facts & Assumptions

Given: The type- $A_{2}$ Coxeter system, its canonical real reflection representation, the positive roots and angular order in the Statement, and the inversion-set map $N$ .

[F1]

For a two-dimensional root plane with $m = 3$ , the canonical angular list has three positive roots (Plane subsystems, their canonical generators, and the angular order of their roots (2)); the extreme-root rays here are $R_{> 0} e_{s}$ and $R_{> 0} e_{t}$ .

[F2]

The reflection subgroup is dihedral of order $6$ (Plane subsystems, their canonical generators, and the angular order of their roots (3)); in this rank-two ambient system $P = V$ and the face point is $x = 0$ , so that subgroup is $W$ .

[F3]

$B (e_{s}, e_{s}) = B (e_{t}, e_{t}) = 1$ and $B (e_{s}, e_{t}) = - \frac{1}{2}$ (The real Coxeter form, its radical, reflections, and form-preserving maps (2)).

[F4]

For $a \in {s, t}$ , $ρ (a) v = v - 2 B (v, e_{a}) e_{a}$ (The real Coxeter form, its radical, reflections, and form-preserving maps (3)).

[F5]

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

[F6]

A finite positive-root set is an inversion set exactly when its restriction to every noncommutative generalized rank-two subsystem is empty, an initial segment, or a final segment (Finite inversion sets are recognized by their rank-two initial or final segments (1)).

[F7]

For a root $α$ , $t_{ρ (w) α} = w t_{α} w^{- 1}$ , and the positive roots are in bijection with reflections (The inversion formula $∣ N (w) ∣ = ℓ (w)$ , the root-reflection dictionary and strong exchange (1)).

[F8]

Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups: the type- $A_{2}$ presentation has $s^{2} = t^{2} = (s t)^{3} = 1$ .

Proof

technique · compute the six group actions on the three positive roots, then check closure and the segment condition directly. This is a finite calculation; no Axiom of Choice (AC) is used

1.1F1F3F4F7F8

Finite setup and root order. From [F8], $s t s = t s t$ ; canceling equal adjacent letters and replacing $s t s t$ by $t s$ and $t s t s$ by $s t$ reduces every word to one of $1, s, t, s t, t s, s t s$ . Hence $W$ is finite and the finite-type suppliers [F1],[F2] apply. Since $m (s, t) = 3$ , [F3, F4] give $ρ (s) e_{s} = - e_{s}$ , $ρ (s) e_{t} = e_{s} + e_{t}$ , $ρ (t) e_{s} = e_{s} + e_{t}$ , and $ρ (t) e_{t} = - e_{t}$ ; by linearity $ρ (s) (e_{s} + e_{t}) = e_{t}$ and $ρ (t) (e_{s} + e_{t}) = e_{s}$ . The orbit definition makes $e_{s} + e_{t}$ a positive root, and [F1] gives exactly three positive roots in the plane. The extreme rays of $V_{+}$ are generated by $e_{s}, e_{t}$ , so their angular order is $(e_{s}, e_{s} + e_{t}, e_{t})$ . By [F7], $t_{e_{s} + e_{t}} = s t_{e_{t}} s = s t s$ , while $t_{e_{s}} = s$ and $t_{e_{t}} = t$ ; hence the reflection order is $(s, s t s, t)$ .

1.2F1

Closure. The roots $e_{s}, e_{t}$ belong to $I$ , and their positive combination $e_{s} + e_{t}$ is a root missing from $I$ , so $I$ is not closed. The complement contains only $e_{s} + e_{t}$ ; the positive-root list has no other root on that ray, so every positive combination of two complement members that is a root is again $e_{s} + e_{t}$ . Hence the complement is closed.

2.1F2F3F4F5

Exhaustive inversion-set calculation. The rank-two presentation gives the six normal forms $1, s, t, s t, t s, s t s$ for $W$ by [F2]. Applying step 1.1 with the rightmost generator acting first, the images of $(e_{s}, e_{s} + e_{t}, e_{t})$ under those elements are $(e_{s}, e_{s} + e_{t}, e_{t})$ , $(- e_{s}, e_{t}, e_{s} + e_{t})$ , $(e_{s} + e_{t}, e_{s}, - e_{t})$ , $(e_{t}, - e_{s}, - e_{s} - e_{t})$ , $(- e_{s} - e_{t}, - e_{t}, e_{s})$ , and $(- e_{t}, - e_{s} - e_{t}, - e_{s})$ , respectively. By [F5], their inversion sets are $\emptyset$ , ${e_{s}}$ , ${e_{t}}$ , ${e_{s} + e_{t}, e_{t}}$ , ${e_{s}, e_{s} + e_{t}}$ , and $Φ_{+}$ . These six elements exhaust $W$ by [F2], and none of these sets is $I$ .

3.1F1F6

Segment criterion. In the order $(e_{s}, e_{s} + e_{t}, e_{t})$ , the initial segments are $\emptyset, {e_{s}}, {e_{s}, e_{s} + e_{t}}, Φ_{+}$ and the final segments are $\emptyset, {e_{t}}, {e_{s} + e_{t}, e_{t}}, Φ_{+}$ . The set $I = {e_{s}, e_{t}}$ is neither. By [F6] it fails the rank-two criterion, agreeing with the exhaustive calculation in step 2.1.

3.2F1F5step 2.1

A valid two-root segment. Step 2.1 gives $N (t s) = {e_{s}, e_{s} + e_{t}} = I^{'}$ , the initial segment consisting of the first two roots in the displayed order.

3.3step 1.2step 2.1

The closed non-inversion set. Step 1.2 proves that $I^{c} = {e_{s} + e_{t}}$ is closed, and the exhaustive list in step 2.1 contains no such singleton inversion set. This proves (v).

4.1F1F2F5F6F7step 1.2step 2.1step 3.1step 3.2step 3.3∎

Conclusion. Steps 1.1-3.3 establish (i)-(v) by finite matrix and set calculations. No witness is selected from an infinite family, so AC is not used.

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Sources

N. Reading and D. E. Speyer, Sortable elements in infinite Coxeter groups, arXiv:0803.2722v3 (2010); Trans. Amer. Math. Soc. 363 (2011) 699-761 (standard reference, not scraped)
A. Bjorner and F. Brenti, Combinatorics of Coxeter Groups, Graduate Texts in Mathematics 231, Springer 2005 (standard reference, not scraped)