TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

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Simple rational representations have a unique highest weight

Statement

Assume the Axiom of Choice inherited from the named suppliers. Let $(V, r)$ be a simple rational representation of a split reductive group $(G, T)$ with Borel subgroup $B \supseteq T$ (Borel subgroups, maximal tori and Borel pairs, Root subgroups of a split reductive group). Then: (a) $V$ contains a primitive vector $v$ , and it is unique up to multiplication by a nonzero scalar; (b) its weight $λ$ is dominant and the weight space $V_{λ} = k v$ is one-dimensional; (c) every weight $μ$ of $V$ satisfies $μ \leq λ$ , i.e. $μ = λ - \sum_{α \in Δ} m_{α} α$ with $m_{α} \geq 0$ ; (d) any two primitive vectors of $V$ have the same weight; this weight is called the highest weight of $V$ .

Facts & Assumptions

Given: A simple rational representation $(V, r)$ of the split reductive group $(G, T)$ with Borel $B \supseteq T$ , unipotent radical $U = B_{u}$ and $B / U ≅ T$ .

[F1]

$B$ is trigonalizable. The split-Borel root-subgroup theorem gives the multiplication map $\prod_{α \in Φ^{+}} U_{α} \to U = B_{u}$ as a $T$ -equivariant isomorphism of varieties for any ordering (Root subgroups of a split reductive group). Since $B$ is split connected solvable with unipotent radical $U$ and $B / U ≅ T$ diagonalizable, $B$ is trigonalizable (Trigonalizable algebraic groups, Unipotent algebraic groups and unipotent representations, Borel subgroups, maximal tori and Borel pairs).

[F2]

Invariant flags. Every finite-dimensional rational representation of a trigonalizable group has a complete flag of subrepresentations (Trigonalizable groups, invariant flags and embeddings into T_n).

[F3]

Finite dimensionality. Simple rational representations of an affine group scheme of finite type are finite-dimensional (Simple rational representations are finite-dimensional).

[F4]

Modules generated by a primitive vector. If a rational representation $W$ of $(G, T)$ is generated as a $G$ -module by a primitive vector of weight $λ$ , then $λ$ is dominant, $W_{λ}$ is one-dimensional, and every weight of $W$ is of the form $λ - \sum_{α \in Δ} m_{α} α$ , $m_{α} \geq 0$ (Modules generated by a primitive vector, Primitive vectors for a Borel pair).

[F5]

The simple roots form a basis of the root lattice, so an expansion in $Δ$ has unique coefficients, and a sum of nonnegative multiples of simple roots equalling zero has all coefficients zero. (Simple roots form a signed integral basis)

Proof

technique · direct

1.1F1F2F3given

By [F1] and [F2], and the finite-dimensionality of $V$ from [F3], the $B$ -module $V$ has a complete flag of subrepresentations; its one-dimensional member is a $B$ -stable line $k v$ with $v \neq 0$ , so $v$ is primitive.

2.1F4step 1.1

Since $V$ is simple and $v \neq 0$ , the $G$ -submodule generated by $v$ is nonzero, hence equal to $V$ ; so $V$ is generated by the primitive vector $v$ . Let $λ$ be its weight. By [F4] the weight $λ$ is dominant, $V_{λ} = k v$ is one-dimensional, and every weight of $V$ has the form $λ - \sum_{α \in Δ} m_{α} α$ with $m_{α} \geq 0$ . This proves (b) and (c).

3.1F4F5step 2.1

Let $v^{'}$ be another primitive vector of $V$ , of weight $λ^{'}$ . Since $V$ is simple, $v^{'}$ also generates $V$ as a $G$ -module, so [F4] applied to $v^{'}$ gives $λ = λ^{'} - \sum_{α \in Δ} m_{α}^{'} α$ with $m_{α}^{'} \geq 0$ , while [F4] applied to $v$ gives $λ^{'} = λ - \sum_{α \in Δ} m_{α} α$ with $m_{α} \geq 0$ . Adding gives $0 = \sum_{α \in Δ} (m_{α} + m_{α}^{'}) α$ , so by [F5] all $m_{α} = m_{α}^{'} = 0$ and $λ = λ^{'}$ .

4.1step 1.1step 2.1step 3.1∎

Any two primitive vectors $v, v^{'}$ have the same weight $λ$ by step 3.1, and since $V_{λ} = k v$ is one-dimensional by step 2.1, $v^{'}$ is a nonzero scalar multiple of $v$ ; this proves (a) and (d). Together with step 2.1 the four assertions hold, and the common weight is the highest weight of $V$ .

Remarks

The primitive vector exists because a split connected solvable group is trigonalizable, so the invariant-flag theorem applies to the finite-dimensional module $V$ .
Uniqueness up to scalar follows from the two inclusions $λ^{'} \leq λ$ and $λ \leq λ^{'}$ together with the one-dimensionality of the top weight space.

Depends on

Used by

Dependency tree · two levels

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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)