LemmaStatement: AI-generatedProof: AI-generated

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Component opposition transports the retained simple roots

Statement

Let $(W, S)$ be a Coxeter system with $S$ finite, in its canonical real representation with simple roots $α_{s}$ , Coxeter form $B$ and contragredient action on $V^{*}$ . For $H \subseteq S$ , write $W_{H} = ⟨ H ⟩$ and $C_{H} = {f \in V^{*} : f (α_{h}) = 0 for h \in H, f (α_{s}) > 0 for s \in S ∖ H}$ . When $W_{H}$ is finite, write $w_{H}$ for its longest element and $ι_{H}$ for its opposition permutation.

Let $I \subseteq S$ , $a \in S ∖ I$ , $L = I + a : = I \cup {a}$ , and $D = [I, a]$ , the vertex set of the connected component containing $a$ in the Coxeter diagram induced on $L$ . Suppose $W_{D}$ is finite. Put $A = D ∖ {a}$ , $b = ι_{D} (a)$ , $K = L ∖ {b}$ , and $v = w_{A} w_{D}$ . Then $v$ maps the set of simple roots indexed by $K$ bijectively to those indexed by $I$ . It fixes $C_{L}$ pointwise. For every $f \in C_{K}$ , $(v f) (α_{a}) = - f (α_{b}) < 0.$ No finiteness assumption on the other components of $L$ is required.

Statement references

Finite parabolic longest elements and disconnected factors

Root signs, inversions and descent translation

Intersections of standard parabolics, the parabolic root subsystem, and global minimality of coset representatives

Real Coxeter representation and contragredient notation

Connected component notation [I,a]

Coxeter diagrams: edges, labels, components and finite type

Facts & Assumptions

[F1]

Finite longest elements are involutions, act on simple roots by negative opposition, and disconnected diagram factors commute, by Finite parabolic longest elements and disconnected factors.

[F2]

The dual action and root-sign convention are those of Root signs, inversions and descent translation.

[F3]

Standard parabolic root subsystems have exactly their supported roots, by Intersections of standard parabolics, the parabolic root subsystem, and global minimality of coset representatives, part (2).

[F4]

The finite-rank canonical representation has $s z = z - 2 B (z, α_{s}) α_{s}$ for $z \in V$ , and its dual action satisfies $(w f) (z) = f (w^{- 1} z)$ . The faces $C_{H}$ have zero coordinates on $H$ and strictly positive coordinates outside $H$ , by Real Coxeter representation and contragredient notation.

[F5]

The set $[I, a]$ is the component containing $a$ in the diagram induced on $I \cup {a}$ ; distinct components have no diagram edges between them, by Connected component notation [I,a].

[F6]

In a finite-rank Coxeter diagram, distinct vertices $s, t$ have an edge exactly when $m (s, t) \geq 3$ , including $m (s, t) = \infty$ ; no edge means $m (s, t) = 2$ . The induced diagram on $L$ is the diagram of the restricted Coxeter matrix, by Coxeter diagrams: edges, labels, components and finite type, Definition (1).

Proof

Given: The finite-rank canonical Coxeter representation, root basis and dual action, with the notation and hypotheses in the Statement. Goal: prove the stated assertions.

Proof technique: direct.

1.1F1F3F4F5F6

Since $A \subseteq D$ , the subgroup $W_{A}$ is finite. By [F1], $w_{D}^{2} = 1$ ; applying it to each simple root gives $ι_{D}^{2} = 1$ , so $ι_{D} (b) = a$ . For $k \in D ∖ {b}$ , opposition gives $w_{D} α_{k} = - α_{ι_{D} (k)}$ , whose index lies in $A$ . Applying $w_{A}$ gives $v α_{k} = α_{ι_{A} (ι_{D} (k))}$ . These indices run bijectively through $A$ , with the restricted root systems understood as in [F3]. Every simple root indexed by $L ∖ D$ is fixed by each generator in $D$ : by [F5] these vertices lie in distinct diagram components, hence by [F6] their Coxeter exponent is two, their off-diagonal form is zero, and the reflection formula in [F4] fixes the root. These roots are fixed by $v$ . Since $K = (D ∖ {b}) \cup (L ∖ D)$ and $I = A \cup (L ∖ D)$ , the claimed simple-root bijection follows.

1.2F2F4

Let $f \in C_{L}$ and $s \in L$ . For every $z \in V$ , the reflection and dual-action formulas give $(s f) (z) = f (s z) = f (z) - 2 B (z, α_{s}) f (α_{s}) = f (z)$ , since $s^{- 1} = s$ and $f (α_{s}) = 0$ . Thus every generator indexed by $L$ fixes $f$ , so every element of $W_{L}$ , in particular $v$ , fixes $C_{L}$ pointwise.

2.1F1F2F4step 1.1∎

By the reflection formula, a generator indexed in $A$ changes only its own root coordinate. Therefore $w_{A} α_{a}$ has coefficient one at $a$ and all other nonzero coordinates in $A$ . By [F1], $v^{- 1} = w_{D} w_{A}$ . Apply $w_{D}$ , which takes each simple root in $D$ to minus its opposition image. The result $v^{- 1} α_{a} = w_{D} w_{A} α_{a}$ is supported on $D$ and has coefficient minus one at $b$ ; its other coordinates are in $D ∖ {b} \subseteq K$ . Since $f \in C_{K}$ vanishes on $K$ and has $f (α_{b}) > 0$ because $b \notin K$ , dual evaluation yields $(v f) (α_{a}) = f (v^{- 1} α_{a}) = - f (α_{b}) < 0$ . This uses finiteness only of $W_{D}$ and its subgroup $W_{A}$ .

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