PropositionStatement: Literature-sourcedProof: 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.

Correspondence between chamber labels and Brink–Howlett morphisms

PDF p. 16: Proposition 4.6

Statement

Let $(W, S)$ be a finite-rank Coxeter system, with vertices $Γ_{0}$ , simple roots $α_{i}$ , length $ℓ$ , and standard parabolic subgroups $W_{I} = ⟨ s_{i} : i \in I ⟩$ . Fix $J \subseteq Γ_{0}$ . A subset $I$ is an associate of $J$ when $W_{I}$ is conjugate to $W_{J}$ in $W$ . The chamber-label set $C h a m (J)$ consists of pairs $(x, I)$ such that $W_{J} x = x W_{I}$ and $x$ has minimum length in $x W_{I}$ . The Brink–Howlett morphisms $I \to K$ are the elements $w \in W$ satisfying $w {α_{k} : k \in K} = {α_{i} : i \in I}$ . The universal wall groupoid has these labels as objects and exactly one arrow from $(x, I)$ to $(y, K)$ , labelled $x^{- 1} y$ .

For associates $I, K$ of $J$ , if $(x, I), (y, K) \in C h a m (J)$ , then $x^{- 1} y \in {Hom}_{B H (J)} (I, K)$ . Conversely, if $w \in {Hom}_{B H (J)} (I, K)$ and $(x, I) \in C h a m (J)$ , then $(x w, K) \in C h a m (J)$ , and $w$ labels the unique arrow from $(x, I)$ to $(x w, K)$ in $\tilde{B H} (J)$ . Under the relative-chamber labelling bijection, the same assertions apply to the corresponding geometric chambers.

Statement references

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

Standard parabolic subgroups and parabolic roots

Algebraic chamber-label set

Brink–Howlett groupoid

Universal wall groupoid on relative chambers

Labelling theorem for relative chambers

Facts & Assumptions

[F1]

A chamber label $(z, L)$ satisfies $W_{J} z = z W_{L}$ and is minimal in $z W_{L}$ . Conversely these two properties define a label. The subgroup $W_{L}$ is generated by the simple reflections indexed by $L$ . Algebraic chamber-label set Standard parabolic subgroups and parabolic roots

[F2]

For each label $(z, L)$ , the element $z$ maps the set of simple roots indexed by $L$ bijectively onto the set indexed by $J$ . Root bases and nonnegative matrices for chamber labels

[F3]

An element $d$ is minimal in $d W_{L}$ exactly when $ℓ (d s_{i}) > ℓ (d)$ for every $i \in L$ ; equivalently, it sends all positive roots of $W_{L}$ to positive roots. Parabolic root subsystems and unique coset minima

[F4]

The geometric representation is faithful, and for each simple generator $s_{i}$ its action is the reflection $v \mapsto v - 2 B (v, α_{i}) α_{i}$ , where $B (α_{i}, α_{i}) = 1$ . These reflections preserve $B$ , hence so does every element of $W$ . A root is positive when its simple-root coordinates are nonnegative. Coxeter-system, roots, length and weak-order conventions Root-sign and inversion-length facts

[F5]

Brink–Howlett arrows are defined by the stated equality of simple-root sets, and the universal wall groupoid has the stated singleton hom-sets. The labelling theorem identifies the algebraic labels with the relative geometric chambers in the finite-rank setting. Brink–Howlett groupoid Universal wall groupoid on relative chambers Labelling theorem for relative chambers

Proof

Given: The finite-rank Coxeter system, subsets and labels in the Statement; neither $W$ nor its standard parabolic subgroups are assumed finite.

Proof technique: direct.

Goal: Prove both implications and the asserted uniqueness of the universal-groupoid arrow.

1.1F2F5

For labels $(x, I)$ and $(y, K)$ , [F2] gives $x {α_{i} : i \in I} = {α_{j} : j \in J} = y {α_{k} : k \in K}$ . Applying $x^{- 1}$ shows that $x^{- 1} y$ sends the simple-root set of $K$ onto that of $I$ . By [F5], it is a Brink–Howlett morphism from $I$ to $K$ .

1.2F1F4F5given

Now let $w : I \to K$ be a Brink–Howlett morphism. For each $k \in K$ , write $w α_{k} = α_{i}$ for its corresponding $i \in I$ . Preservation of $B$ and the reflection formula in [F4] show that $w s_{k} w^{- 1}$ and $s_{i}$ have the same action on every vector; faithfulness makes them equal in $W$ . Since $w$ bijects the two simple-root sets, these equalities give $w W_{K} w^{- 1} = W_{I}$ . Therefore, using $W_{J} x = x W_{I}$ , we obtain $W_{J} x w = x w W_{K}$ .

1.3F1F3F4F5given

By minimality of $x$ in $x W_{I}$ , [F3] makes each $x α_{i}$ with $i \in I$ positive. For every $k \in K$ , the root $x w α_{k}$ is one of these positive roots. Let $β$ be any positive root of $W_{K}$ . The reflection formula in [F4] shows that each $s_{k}$ with $k \in K$ preserves $V_{K} = {span}_{R} {α_{k} : k \in K}$ . By the definition of parabolic roots, $β$ therefore lies in $V_{K}$ ; its positivity gives $β = \sum_{k \in K} c_{k} α_{k}$ with $c_{k} \geq 0$ . By linearity, $x w β = \sum_{k \in K} c_{k} (x w α_{k})$ has nonnegative simple-root coordinates. It is a root, since $x w \in W$ , and hence is positive by [F4]. Thus $x w$ sends every positive root of $W_{K}$ to a positive root.

2.1F1F3F5step 1.2step 1.3∎

Apply the all-positive-parabolic-roots criterion in [F3] to step 1.3: $x w$ is minimal in $x w W_{K}$ . Together with step 1.2, [F1] gives $(x w, K) \in C h a m (J)$ . The singleton arrow from $(x, I)$ to $(x w, K)$ has label $x^{- 1} (x w) = w$ . Finally, [F5] transfers these assertions to the corresponding geometric chambers.

Source notes

The proof follows the article's reduction of the first assertion to the simple-root sets associated with individual labels. The converse uses the same positivity and coset-minimality mechanism, with the conjugation of standard parabolic subgroups made explicit. The geometric identification used here is supplied only in finite rank; no infinite-rank topological extension is asserted.

Depends on

Used by · 4

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 16, Proposition 4.6 and proof