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

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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.

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Verma-flag multiplicities are independent of the flag

Statement

Assume the Axiom of Choice (The Axiom of Choice). If $X \in O$ admits two finite Verma flags with corresponding multiplicities $m_{μ}$ and $m_{μ}^{'}$ (Finite Verma flags and their multiplicities), then $m_{μ} = m_{μ}^{'}$ for every weight $μ$ . Hence the multiplicity $(X : Δ (μ))$ of Finite Verma flags and their multiplicities is well defined.

Facts & Assumptions

Given: The Axiom of Choice, an object $X$ with two finite Verma flags and their multiplicity functions $m, m^{'}$ .

[F1]

If $0 = X_{0} \subseteq X_{1} \subseteq \dots \subseteq X_{n} = X$ is a Verma flag with factors $X_{i} / X_{i - 1} ≅ Δ (μ_{i})$ , then $[X] = \sum_{i = 1}^{n} [Δ (μ_{i})]$ in the Grothendieck group, where $[Δ (μ)] = [M (μ)]$ , and $m_{μ} = # {i : μ_{i} = μ}$ is the multiplicity; all but finitely many $m_{μ}$ vanish (Finite Verma flags and their multiplicities, The Grothendieck group and character of O).

[F2]

The classes $[M (λ)]$ , equivalently the classes $[Δ (λ)]$ , form a $Z$ -basis of $K_{0} (O)$ (Simple and standard bases of K0(O)).

Proof

technique · direct comparison of two basis expansions in the Grothendieck group

1.1F1given

The two flags give two finite expansions of the same class, $[X] = \sum_{μ} m_{μ} [Δ (μ)]$ and $[X] = \sum_{μ} m_{μ}^{'} [Δ (μ)]$ , in $K_{0} (O)$ .

2.1F2step 1.1∎

Since the standard classes $[Δ (μ)] = [M (μ)]$ form a $Z$ -basis of $K_{0} (O)$ , the coefficient of each basis element in a class is uniquely determined. Comparing the two expansions of $[X]$ from step 1.1 therefore gives $m_{μ} = m_{μ}^{'}$ for every weight $μ$ , so the multiplicity $(X : Δ (μ))$ is independent of the chosen flag.

Depends on

Used by

Verma filtrations are not closed under quotients Counterexample
Translation through the sl2 wall Example

Cited to discharge well-definedness by Finite Verma flags and their multiplicities.

Dependency tree · two levels

17 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

Lin Chen, lecture notes (Spring 2024), Lecture 9, Proposition-Definition 2.1 (standard reference, not scraped)
Pavel Etingof, Representations of Lie Groups (18.757, Fall 2023), Sec. 23.1 (standard reference, not scraped)