LemmaStatement: Literature-sourcedProof: 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.

Simple reflections preserve weight multiplicities

Statement

Assume the Axiom of Choice. Let $g$ be a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $h$ and root system $Φ$ , choose a positive system $Φ^{+}$ with base $Δ = {α_{1}, \dots, α_{r}}$ (Positive systems and simple roots), let $V$ be a finite-dimensional representation of $g$ , and let $W$ be the Weyl group of $Φ$ , acting on $h^{*}$ by complex-linear extension of its action on $E = {span}_{R} Φ$ (Weyl group, The roots form a reduced crystallographic Euclidean root system). Then for every simple root $α_{i}$ and every $μ \in h^{*}$ $\dim V_{μ} = \dim V_{s_{i} (μ)}, s_{i} (μ) = μ - μ (h_{α_{i}}) α_{i},$ and consequently $\dim V_{w μ} = \dim V_{μ}$ for every $w \in W$ and every $μ \in h^{*}$ (Weight and weight space).

Facts & Assumptions

Given: The Axiom of Choice, such $g, h, Φ$ , the chosen positive system $Φ^{+}$ with simple roots $Δ$ , a finite-dimensional representation $V$ , and the Weyl group $W$ acting on $h^{*}$ .

[A1]

The Axiom of Choice is assumed; it enters through the root-space theory supplying [L1] and the abstract root-system identification [L4] (The Axiom of Choice).

[L1]

For every root $α$ the coroot $h_{α}$ and suitable $e_{α} \in g_{α}$ , $f_{α} \in g_{- α}$ form a copy of ${s l}_{2}$ with $[e_{α}, f_{α}] = h_{α}$ ; moreover $α (h_{α}) = 2$ and $s_{α} (μ) = μ - μ (h_{α}) α$ is the reflection of Root reflection defined by a coroot (The root sl_2 triple, Coroot of a Lie-algebra root).

[L2]

A finite-dimensional ${s l}_{2}$ -module is a direct sum of irreducible submodules, and on each irreducible summand the operators $e$ and $f$ move along a finite weight string; in particular they act nilpotently (Finite-dimensional representations of sl_2).

[L3]

For $μ \in h^{*}$ , the weight space is $V_{μ} = {v : H \cdot v = μ (H) v for all H \in h}$ (Weight and weight space).

[L4]

The roots of $g$ form a reduced crystallographic Euclidean root system on $E$ , and its root reflections coincide with the $s_{α}$ of [L1] after complex-linear extension to $h^{*}$ (The roots form a reduced crystallographic Euclidean root system, Weyl group, Root reflection defined by a coroot).

[L5]

Relative to the chosen base $Δ$ , every element of the Weyl group is a product of the corresponding simple reflections (Positive systems and simple roots, Weyl length equals inversion number).

Proof

technique · direct

1.1

Fix a simple root $α = α_{i}$ and the ${s l}_{2}$ -triple $(e_{α}, f_{α}, h_{α})$ of [L1], and write $ρ$ for the action of $g$ on $V$ . By [L2], the endomorphisms $E = ρ (e_{α})$ and $F = ρ (f_{α})$ are nilpotent. Hence the finite sums $\exp (E)$ and $\exp (- F)$ are defined and invertible, and so is $N_{α} = \exp (E) \exp (- F) \exp (E)$ .

A1L1L2algebra

2.1

Let $H \in h$ and put $a = α (H)$ . In the adjoint action of the root triple, the relations of [L1] give $\exp (ad e_{α}) H = H - a e_{α}$ , then $\exp (- ad f_{α}) (H - a e_{α}) = H - a e_{α} - a h_{α}$ , and applying $\exp (ad e_{α})$ once more gives $H - a h_{α}$ . Conjugation by an exponential satisfies $\exp (E) ρ (x) \exp (- E) = ρ (\exp (ad e_{α}) x)$ , with finite series here. Therefore $N_{α} ρ (H) N_{α}^{- 1} = ρ (H - α (H) h_{α})$ .

L1step 1.1algebra

3.1

Since the reflection $r_{α} (H) = H - α (H) h_{α}$ is an involution, step 2.1 also gives $N_{α}^{- 1} ρ (H) N_{α} = ρ (r_{α} (H))$ . If $v \in V_{μ}$ , then for every $H \in h$ one has $ρ (H) N_{α} v = N_{α} ρ (r_{α} (H)) v = μ (r_{α} (H)) N_{α} v = s_{α} (μ) (H) N_{α} v$ . Thus $N_{α} (V_{μ}) \subseteq V_{s_{α} (μ)}$ . Applying the same argument to $N_{α}^{- 1}$ gives the reverse inclusion, so $N_{α}$ restricts to an isomorphism $V_{μ} ≅ V_{s_{α} (μ)}$ . Hence $\dim V_{μ} = \dim V_{s_{α} (μ)}$ for every $μ \in h^{*}$ .

L1L3L4step 2.1algebra

4.1

Every $w \in W$ is a product of simple reflections by [L5]. Applying step 3.1 successively to those factors gives $\dim V_{w μ} = \dim V_{μ}$ for every $w \in W$ and $μ \in h^{*}$ .

L4L5step 3.1

5.1

The stated equalities follow from steps 3.1 and 4.1.

step 3.1step 4.1∎

Depends on

Used by

Extremal Weyl-orbit weights Proposition
Highest weight of the dual representation Proposition

Dependency tree · two levels

45 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. (standard reference, not scraped)
Alexander Kirillov Jr., An Introduction to Lie Groups and Lie Algebras (standard reference, not scraped)