LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

The normalizer of the torus permutes weight spaces

Statement

Assume the Axiom of Choice inherited from the named suppliers. Let $(V, r)$ be a rational representation of a split reductive group $(G, T)$ and let $n \in N_{G} (T) (R)$ for a $k$ -algebra $R$ . If $v \in V_{λ}$ , then $n \cdot (v \otimes 1) \in (V \otimes_{k} R)_{n λ}$ , where $n λ : T_{R} \to G_{m, R}$ is the character $t \mapsto λ (n^{- 1} t n)$ , tested over every $R$ -algebra. If $R$ is disconnected, this character can vary between components; the target denotes its eigensubmodule over $R$ . Consequently, for every $w \in W (G, T)$ the weight spaces $V_{λ}$ and $V_{w λ}$ have the same dimension, and the set of weights of $(V, r)$ is stable under $W$ (The Weyl group, Borel subgroups and chambers, The root datum of a split reductive group).

Facts & Assumptions

Given: A rational representation $(V, r)$ of the split reductive group $(G, T)$ , a weight vector $v \in V_{λ}$ and a point $n \in N_{G} (T) (R)$ for a $k$ -algebra $R$ .

[F1]

The character $n λ$ . For $n \in N_{G} (T) (R)$ the assignment $t \mapsto λ (n^{- 1} t n)$ is a character $n λ \in X (T)_{R}$ of $T_{R}$ : it is a morphism in $t$ and multiplicative because conjugation is a group homomorphism (The root datum of a split reductive group, Weights, dominant weights and the highest-weight order of a rational representation).

[F2]

Weight vectors. For $t \in T (R)$ and $v \in V_{λ}$ one has $t \cdot v = λ (t) v$ , and the weight spaces are the eigenspaces of the $T$ -action (Weights, dominant weights and the highest-weight order of a rational representation).

[F3]

Weyl group. $W (G, T) = N_{G} (T) (k) / T (k)$ , and every $w \in W$ has representatives $n \in N_{G} (T) (k)$ and $n^{- 1}$ for $w^{- 1}$ (The Weyl group, Borel subgroups and chambers, The root datum of a split reductive group).

Proof

technique · direct

1.1F1F2given

For $t \in T (R)$ one computes $t \cdot (n \cdot v) = n \cdot ((n^{- 1} t n) \cdot v) = n \cdot (λ (n^{- 1} t n) v) = λ (n^{- 1} t n) (n \cdot v) = (n λ) (t) (n \cdot v)$ in $V_{R}$ : the first equality uses that the action is a left action and $n^{- 1} t n \in T (R)$ , and the second uses [F2] and $R$ -linearity of the action of $n$ . Hence $n \cdot (v \otimes 1)$ belongs to the stated eigensubmodule of $V \otimes_{k} R$ , with the calculation valid over every $R$ -algebra.

2.1F3step 1.1

Applying step 1.1 to $n \in N_{G} (T) (R)$ gives a bijection $V_{λ} \otimes_{k} R \to (V \otimes_{k} R)_{n λ}$ , $v \mapsto n \cdot v$ , with inverse given by $n^{- 1}$ ; taking $R = k$ and representatives of $w$ shows $\dim_{k} V_{λ} = \dim_{k} V_{w λ}$ for every $w \in W$ .

3.1step 2.1

Since $V_{χ} \neq 0$ exactly for the weights of $V$ , step 2.1 shows that the set of weights is stable under the action $w ⁣ : λ \mapsto w λ$ of $W$ .

4.1step 1.1step 2.1step 3.1∎

Steps 1.1, 2.1 and 3.1 establish the three assertions.

Depends on

Used by

Dependency tree · two levels

23 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

J. S. Milne, Algebraic Groups (corrected 2022 printing, Cambridge University Press) (standard reference, not scraped)
Robert Steinberg, Lectures on Chevalley Groups (Yale University, 1967; notes prepared by J. Faulkner and R. Wilson) (standard reference, not scraped)