ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedaudited 2026-09-22

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.

Weyl groups of B_n and D_n

Example

For $n \geq 1$ , $W (B_{n})$ is the full group of signed permutations of the coordinates, of order $2^{n} n!$ . For $n \geq 2$ , $W (D_{n})$ is the subgroup of signed permutations with an even number of sign changes, of order $2^{n - 1} n!$ .

Facts & Assumptions

Given: The coordinate model $B_{n} = {\pm ε_{i}} \cup {\pm ε_{i} \pm ε_{j}}$ in $R^{n}$ for $n \geq 1$ , and the coordinate model $D_{n} = {\pm ε_{i} \pm ε_{j}}$ in $R^{n}$ for $n \geq 2$ .

[L1]

The coordinate sets in the Given data are the classical reduced crystallographic root systems, including $B_{1} = A_{1}$ and $D_{2} = A_{1} ⊔ A_{1}$ (Root systems of the classical complex Lie algebras).

[L2]

For a root $α$ , $s_{α} (x) = x - 2 (x, α) α / (α, α)$ , and the Weyl group is generated by these reflections (Weyl group).

Verification

technique · direct

1.1

Substituting the orthonormal coordinate vectors into [L2] shows that $s_{ε_{i}}$ negates coordinate $i$ , $s_{ε_{i} - ε_{j}}$ exchanges coordinates $i, j$ , and $s_{ε_{i} + ε_{j}}$ sends $(x_{i}, x_{j})$ to $(- x_{j}, - x_{i})$ , fixing all other coordinates. Negating the root leaves the reflection unchanged. Thus every reflection of either system is a signed permutation.

L1L2algebra

2.1

A signed permutation is uniquely specified by a permutation of the coordinate axes and a sign on each image axis. In type $B_{n}$ , all individual sign changes occur by step 1.1, as do all transpositions. Transpositions generate every permutation (move the desired entry to each position successively), so these reflections generate every signed permutation. Conversely every generating root reflection is such a permutation. Hence $W (B_{n})$ is exactly the full signed permutation group.

L1L2step 1.1algebra

2.2

For a signed permutation define its sign parity as the product of its $n$ signs. Under composition this product multiplies, because permuting signs does not change their product. Each $D_{n}$ root reflection has sign parity $+ 1$ . Conversely, the product $s_{ε_{i} + ε_{j}} s_{ε_{i} - ε_{j}}$ negates exactly coordinates $i, j$ and fixes the rest, by step 1.1. Any even set of coordinates can be partitioned into pairs, so products of these two-reflection operations realize every even sign pattern. The difference-root reflections also generate every permutation. Hence all and only signed permutations with an even number of negative signs occur in $W (D_{n})$ . This uses a pair of different types of reflections, rather than the unsupported assertion that an even number of sum-root reflections alone produces every even pattern.

L1L2step 1.1algebra

3.1

There are $n!$ permutations and $2^{n}$ sign choices, independently, so $∣ W (B_{n}) ∣ = 2^{n} n!$ . For $D_{n}$ , the first $n - 1$ signs are arbitrary and the last is forced by their product, giving $2^{n - 1} n!$ . At $n = 1$ the $B_{1}$ group is ${1, - 1}$ of order two. At $n = 2$ the $D_{2}$ group consists of the identity and coordinate swap, each with either both signs positive or both negative, of order four. No assertion is made for the excluded $D_{0}, D_{1}$ .

step 2.1step 2.2algebra∎

Depends on

Used by

Restricted root systems may be nonreduced Proposition

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Chapter II (standard reference, not scraped)