LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30

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Complex Specht modules have nondegenerate Hermitian self-pairing

Statement

For every $λ ⊢ n$ , $S^{λ}$ is nonzero and $S^{λ} \cap (S^{λ})^{⊥} = {0}$ for the positive definite invariant Hermitian tabloid product.

Facts & Assumptions

Given: $n \geq 0$ and $λ ⊢ n$ .

[F1]

The tabloid-basis Hermitian product is positive definite: $⟨ x, x ⟩ = \sum_{T} ∣ a_{T} ∣^{2} > 0$ whenever $x = \sum_{T} a_{T} T \neq 0$ (Invariant Hermitian product on a tabloid module).

[F2]

$S^{λ}$ is the complex span of the polytabloids $e_{s}$ (Column antisymmetrizers, polytabloids, and Specht modules).

[F3]

For every tableau $t$ , the coefficient of ${t}$ in $e_{t}$ is $1$ (Column antisymmetrizers, polytabloids, and Specht modules).

[F4]

For every partition, the canonical row-filled $λ$ -tableau exists (Young subgroups, tabloids, and permutation modules).

[F5]

The orthogonal complement is defined by vanishing of the inner product against every vector of the subspace (Orthogonality and the orthogonal complement).

Proof

technique · direct

1.1givenF2F3F4algebra

If $λ = \emptyset$ , take its empty tableau; otherwise take the canonical row-filled tableau $t_{0}$ from [F4]. By [F2], $e_{t_{0}} \in S^{λ}$ , and by [F3] its coefficient at ${t_{0}}$ is $1$ . Thus $e_{t_{0}} \neq 0$ and $S^{λ} \neq 0$ .

2.1givenF1F2F5algebra∎

Let $v \in S^{λ} \cap (S^{λ})^{⊥}$ . By [F5], $⟨ v, w ⟩ = 0$ for every $w \in S^{λ}$ ; taking $w = v$ gives $⟨ v, v ⟩ = 0$ . Positive definiteness [F1] implies $v = 0$ . The zero vector belongs to both spaces by [F2] and [F5], so $S^{λ} \cap (S^{λ})^{⊥} = {0}$ .

Depends on

Used by

Complex Specht modules are irreducible Theorem

Dependency tree · two levels

12 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

Charlotte Chan, Representation Theory of Symmetric Groups, Chapter 9, Definition 9.1, Remark 9.2 and Lemma 9.3, printed pp. 31-32; her form is bilinear, and the local form is Hermitian (standard reference, not scraped)
David A. Craven, Groups, Geometries and Representation Theory, Section 2.1, Corollary 2.4 and its characteristic-zero conclusion, printed p. 20; the local proof uses Hermitian positivity (standard reference, not scraped)