LemmaStatement: 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.

Root bases and nonnegative matrices for chamber labels

PDF p. 16: Proposition 4.6(1), proof

Statement

Let $(W, S)$ be a finite-rank Coxeter system with vertex set $Γ_{0}$ , canonical geometric representation $V = ⨁_{i \in Γ_{0}} R α_{i}$ , and positive roots $Φ^{+}$ . For $I \subseteq Γ_{0}$ , put $V_{I} = {span}_{R} {α_{i} : i \in I}$ and let $Φ_{I}^{+}$ be the positive roots of the standard parabolic subgroup $W_{I}$ . Fix $J \subseteq Γ_{0}$ and $(y, K) \in C h a m (J)$ ; explicitly, $W_{J} y = y W_{K}$ and $y$ has minimum length in $y W_{K}$ .

Then $y^{- 1} α_{j} \in Φ_{K}^{+}$ for every $j \in J$ , and $y α_{k} \in Φ_{J}^{+}$ for every $k \in K$ . The maps $y : V_{K} \to V_{J}$ and $y^{- 1} : V_{J} \to V_{K}$ are inverse isomorphisms. Their matrices in the indicated simple-root bases are entrywise nonnegative and are permutation matrices. Consequently $y {α_{k} : k \in K} = {α_{j} : j \in J}$ . The assertions include empty $J$ and $K$ ; neither $W$ nor the parabolic subgroups are assumed finite.

Statement references

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

Algebraic chamber-label set

Standard parabolic subgroups and parabolic roots

Coordinate columns [v]_B and matrices [T]_B^C of linear maps relative to ordered bases

Facts & Assumptions

[F1]

The geometric representation has symmetric bilinear form $B$ with $B (α_{i}, α_{i}) = 1$ and $s_{i} (v) = v - 2 B (v, α_{i}) α_{i}$ . Its roots are the $W$ -translates of simple roots, and positivity means nonnegative simple-root coordinates. Coxeter-system, roots, length and weak-order conventions

[F2]

The label condition is $W_{J} y = y W_{K}$ , with $y$ of minimum length in this common set. The standard parabolic $W_{I}$ is generated by the $s_{i}$ for $i \in I$ , and $Φ_{I}$ consists of their translates of the simple roots indexed by $I$ . Algebraic chamber-label set Standard parabolic subgroups and parabolic roots

[F3]

For finite rank, $Φ \cap V_{I} = Φ_{I}$ . A minimum $d$ in $d W_{I}$ sends $Φ_{I}^{+}$ into $Φ^{+}$ , and a minimum $d$ in $W_{I} d$ has $d^{- 1} Φ_{I}^{+} \subseteq Φ^{+}$ . Parabolic root subsystems and unique coset minima

[F4]

An invertible real square matrix whose entries and inverse entries are nonnegative is a product of a permutation matrix and a strictly positive diagonal matrix; the empty matrix is included. Nonnegative inverse matrices preserve extremal basis rays

[F5]

Relative to ordered domain and codomain bases, the column indexed by a domain basis vector is the coordinate column of its image under the linear map; empty bases give the corresponding zero-sized matrices. Coordinate columns [v]_B and matrices [T]_B^C of linear maps relative to ordered bases

Proof

Given: The finite-rank Coxeter system, $J$ , and label $(y, K)$ in the Statement.

Proof technique: direct.

Goal: Prove the two positive parabolic-root inclusions and the permutation-matrix conclusion.

1.1F1F2

For any subset $I$ , each generator $s_{i}$ with $i \in I$ acts as the identity modulo $V_{I}$ , by the reflection formula. The same is therefore true of every element of $W_{I}$ . From $W_{J} y = y W_{K}$ , we have $y^{- 1} W_{J} y = W_{K}$ and $y W_{K} y^{- 1} = W_{J}$ . Fix $j \in J$ and put $β = y^{- 1} α_{j}$ . The element $y^{- 1} s_{j} y \in W_{K}$ sends $β$ to $- β$ ; since it is the identity modulo $V_{K}$ , its difference on $β$ shows $2 β \in V_{K}$ . Thus $y^{- 1} V_{J} \subseteq V_{K}$ . Interchanging $J$ and $K$ and replacing $y$ by $y^{- 1}$ gives $y V_{K} \subseteq V_{J}$ . Applying $y$ to the first inclusion shows that both inclusions are equalities, so the restrictions of $y$ and $y^{- 1}$ are inverse isomorphisms.

2.1F1F2F3F5step 1.1

Each $y^{- 1} α_{j}$ is a root in $V_{K}$ , hence belongs to $Φ_{K}$ by [F3]. Since $y$ is minimum in $W_{J} y$ , [F3] also makes it positive. Similarly, each $y α_{k}$ is a root in $V_{J}$ and is positive because $y$ is minimum in $y W_{K}$ . This proves both parabolic-root inclusions. Their nonnegative simple-root coordinates are precisely the columns of the respective matrices of the inverse isomorphisms in 1.1.

3.1F1F4F5step 1.1step 2.1∎

Choose orderings of the two finite simple-root bases. Their cardinalities agree by 1.1, and the two matrices are inverse and entrywise nonnegative by steps 1.1 and 2.1. By [F4], the matrix of $y$ is a permutation matrix times a strictly positive diagonal matrix. Thus each $y α_{k}$ is a positive scalar multiple $c α_{j}$ of one simple root, with the indices matched by a permutation. The reflection formula and $B (α_{i}, α_{i}) = 1$ show by expansion that each $s_{i}$ preserves $B$ ; hence every element of $W$ preserves $B$ , and every root has self-pairing $1$ . Consequently $1 = B (y α_{k}, y α_{k}) = c^{2}$ , so $c = 1$ . The matrix of $y$ is therefore a permutation matrix, as is its inverse, and the displayed equality of simple-root sets follows. For empty bases the same conclusion is vacuous and the matrices are the unique $0 \times 0$ matrices.

Depends on

Used by · 1

Correspondence between chamber labels and Brink–Howlett morphisms

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(1), proof