PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedaudited 2026-09-22

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The highest-weight space is one-dimensional

Statement

Assume the Axiom of Choice. Let $g$ be a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $h$ and a chosen positive system, and let $V$ be a finite-dimensional irreducible highest weight module of highest weight $λ$ (Highest-weight vectors and modules). Then the $λ$ -weight space of $V$ is one-dimensional: $\dim V_{λ} = 1.$

Facts & Assumptions

Given: The Axiom of Choice, such $g, h$ , a chosen positive system, and a finite-dimensional irreducible highest weight module $V$ of highest weight $λ$ .

[A1]

The Axiom of Choice is assumed; it enters through the root-space theory used by the cited suppliers (The Axiom of Choice).

[L1]

There is a highest weight vector $v \in V$ of weight $λ$ generating $V$ as a $g$ -module, and $V$ is irreducible (Highest-weight vectors and modules, Weight and weight space).

[L2]

The subrepresentation of an irreducible module generated by any highest weight vector is the whole module: here $U (g) v = V$ (An irreducible module is generated by its highest-weight vector).

[L3]

A module generated by a highest weight vector $v$ of weight $λ$ satisfies $V = U (n^{-}) C v$ and has $λ$ -weight space exactly $C v$ (Highest weight modules lie below the top weight).

Proof

technique · direct

1.1

Take a highest weight vector $v$ of weight $λ$ generating $V$ , as [L1] provides.

A1L1

1.2

By [L2], $U (g) v = V$ , and by the definition of a highest weight module this is the statement that $v$ generates $V$ ; hence the pair $(V, v)$ satisfies the hypothesis of the weight bound [L3].

L1L2

2.1

Applying [L3] to $(V, v)$ gives $V_{λ} = C v$ .

L3step 1.2

3.1

Therefore $\dim V_{λ} = 1$ , which is the assertion.

step 2.1∎

Depends on

Used by

Extremal Weyl-orbit weights Proposition
Highest weight of the dual representation Proposition
Top summand in a tensor product Proposition
Simple highest-weight modules are classified by highest weight Theorem

Dependency tree · two levels

30 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)