LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Componentwise crossings are inverse elements

Statement

Let $I, a, D, A, b, K, v$ have the notation and finite-component hypothesis of component opposition. Then $[K, b] = D$ , $ι_{D} (b) = a$ , and $w_{D ∖ {b}} w_{D} = v^{- 1} .$ Thus the two componentwise crossing elements multiply to the identity in either order.

Facts & Assumptions

[F1]

The notation, finite-component hypothesis and simple-root bijection are supplied by Component opposition transports the retained simple roots.

[F2]

Longest elements and opposition are as in Finite parabolic longest elements and disconnected factors.

[F3]

For a subset $H \subseteq S$ and $t \in S ∖ H$ , $[H, t]$ is the vertex set of the connected component containing $t$ in the Coxeter diagram induced on $H \cup {t}$ , by Connected component notation [I,a].

[F4]

For every $H \subseteq S$ , $(W_{H}, H)$ is a Coxeter system and its intrinsic length is the restriction of the ambient length, by Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification, part (2).

[F5]

In a finite Coxeter system, its longest element $w_{0}$ conjugates each simple generator $s$ to $σ (s)$ , where $w_{0}$ takes the simple root indexed by $s$ to the negative simple root indexed by $σ (s)$ , by The longest element as the opposition of the chamber, and longest elements of finite parabolics, part (1)(v).

[F6]

In the canonical real representation, $s z = z - 2 B (z, α_{s}) α_{s}$ , where the Coxeter form on the simple-root basis is determined by the Coxeter matrix, by Real Coxeter representation and contragredient notation.

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.

1.1F1F2F3

By [F1], $b = ι_{D} (a) \in D \subseteq L$ and $K = L ∖ {b}$ , so $b \notin K$ and $K + b = L = I + a$ . By [F3], both $[I, a]$ and $[K, b]$ refer to components of this same induced diagram. Its component $D = [I, a]$ contains $b$ , so $[K, b] = D$ . By [F2], $w_{D}^{2} = 1$ and $w_{D} α_{d} = - α_{ι_{D} (d)}$ for $d \in D$ . Applying $w_{D}^{2}$ to the simple-root basis gives $ι_{D}^{2} = 1$ , hence $ι_{D} (b) = a$ .

2.1F1F2F4F5F6step 1.1∎

Put $B = D ∖ {b}$ . By [F4], $(W_{D}, D)$ is a Coxeter system; it is finite by [F1]. Its canonical representation on the span of the simple roots in $D$ is the restriction of the ambient canonical representation: by [F6], on each generator and basis root the defining reflection formula uses precisely the restricted Coxeter matrix. Thus the permutation in [F5] is the same $ι_{D}$ as in [F2], and [F5] gives $w_{D} s w_{D} = ι_{D} (s)$ for $s \in D$ . Since $ι_{D}$ is a permutation with $ι_{D} (a) = b$ , conjugation by $w_{D}$ bijects $A = D ∖ {a}$ with $B$ . It therefore restricts to an isomorphism $W_{A} \to W_{B}$ preserving intrinsic word lengths: a word in $A$ maps to a word of the same length in $B$ , and conjugating back gives the reverse length inequality. Both groups are finite subgroups of $W_{D}$ , and [F4] identifies their intrinsic lengths with the lengths used for their unique longest elements in [F2]. Consequently $w_{B} = w_{D} w_{A} w_{D}$ . By [F2] both $w_{A}$ and $w_{D}$ are involutions, so $w_{B} w_{D} = w_{D} w_{A} = (w_{A} w_{D})^{- 1} = v^{- 1}$ . This proves the displayed equality and both identity products.

Depends on

Used by · 2

Dependency tree · 2 levels

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