DefinitionDefinition: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-02

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.

Integral tabloid form and Specht Gram matrix

Definition

Let $n \geq 0$ and $λ ⊢ n$ , and keep the integral tabloid module $M_{Z}^{λ}$ and the integral Specht lattice $S_{Z}^{λ}$ with its standard basis (Integral Specht lattice and base change). The integral tabloid form is the $Z$ -bilinear form

$β : M_{Z}^{λ} \times M_{Z}^{λ} ⟶ Z, β (\sum_{T} a_{T} T, \sum_{U} b_{U} U) : = \sum_{T \in Ω_{λ}} a_{T} b_{T},$

the unique $Z$ -bilinear form for which the tabloid basis is orthonormal: $β (T, U) = δ_{T U}$ for all $λ$ -tabloids $T, U$ . It is symmetric because $δ_{T U} = δ_{U T}$ , and nondegenerate because $β (x, U) = 0$ for all $U$ forces every tabloid coefficient of $x$ to vanish.

Let $s_{1}, \dots, s_{d}$ be the standard $λ$ -tableaux and $e_{s_{1}}, \dots, e_{s_{d}}$ the corresponding standard polytabloids, a $Z$ -basis of $S_{Z}^{λ}$ . The integral Specht Gram matrix is

$G_{λ} : = (β (e_{s_{i}}, e_{s_{j}}))_{1 \leq i, j \leq d} \in M_{d} (Z),$

the matrix of the restriction $β ∣_{S_{Z}^{λ}}$ in the standard basis; its entries are integers because each $e_{s}$ has tabloid coefficients in ${0, 1, - 1}$ . For a commutative ring $R$ let $β_{R} : M_{R}^{λ} \times M_{R}^{λ} \to R$ be the $R$ -bilinear form with $β_{R} (T, U) = δ_{T U}$ on the tabloid basis of $M_{R}^{λ}$ . Under the canonical identification $M_{R}^{λ} ≅ R \otimes_{Z} M_{Z}^{λ}$ the form $β_{R}$ is the scalar extension of $β$ , $β_{R} (1 \otimes x, 1 \otimes y) = 1 \otimes β (x, y)$ , and the Gram matrix of $β_{R}$ in the standard basis $1 \otimes e_{s_{i}}$ of $S_{R}^{λ}$ is the scalar extension of $G_{λ}$ to $R$ .

Two warnings are built into the definition. First, $β$ is a symmetric bilinear form, not the complex Hermitian form of Invariant Hermitian product on a tabloid module: the latter is conjugate-linear in its first variable and positive definite, and neither its conjugation nor its positivity has a meaning over a general commutative ring. Positivity of $β$ is asserted only after embedding in $R$ or $C$ : for real coefficients $β (x, x) = \sum_{T} a_{T}^{2} \geq 0$ with equality only for $x = 0$ . Second, the reduction of $G_{λ}$ modulo a prime is governed by the divisibility theory of its entries, and cannot be decided from the positive-definiteness of the Hermitian form.

The form satisfies $β (σ x, σ y) = β (x, y)$ for all $σ \in S_{n}$ , so every $κ_{t}$ is self-adjoint for $β$ : $β (κ_{t} x, y) = β (x, κ_{t} y)$ .

Facts & Assumptions

Given: An integer $n \geq 0$ , a partition $λ ⊢ n$ , and the definitions above.

[F1]

$M_{Z}^{λ}$ is free with the tabloids as $Z$ -basis, so every element has a unique finite integer coordinate expression; for every commutative ring $R$ there is a natural identification $M_{R}^{λ} ≅ R \otimes_{Z} M_{Z}^{λ}$ (Integral Specht lattice and base change).

[F2]

The standard polytabloids form a $Z$ -basis of $S_{Z}^{λ}$ (Integral Specht lattice and base change).

[F3]

The tabloids form a basis of the permutation module $M^{λ}$ and each $σ \in S_{n}$ permutes the tabloids bijectively (Young subgroups, tabloids, and permutation modules).

[F4]

$κ_{t} = \sum_{γ \in C_{t}} sgn (γ) γ$ and $e_{t} = \sum_{γ \in C_{t}} sgn (γ) {γ \cdot t}$ , so all tabloid coefficients of $e_{t}$ lie in ${0, 1, - 1}$ (Column antisymmetrizers, polytabloids, and Specht modules).

[F5]

The published Hermitian form on the complex tabloid module is conjugate-linear in its first variable, positive definite, invariant under the unitary action, and every $κ_{t}$ is self-adjoint for it (Invariant Hermitian product on a tabloid module).

Proof

technique · direct

1.1givenF1algebra

The prescription $β (T, U) = δ_{T U}$ on the tabloid basis extends uniquely to a $Z$ -bilinear map by [F1], and the formula $β (\sum_{T} a_{T} T, \sum_{U} b_{U} U) = \sum_{T} a_{T} b_{T}$ is finite because only finitely many coefficients are nonzero. If $β (x, U) = 0$ for every tabloid $U$ , then taking $U$ to be a tabloid occurring in $x$ gives $a_{U} = 0$ , so all coefficients vanish and $x = 0$ ; the form is nondegenerate.

1.2givenF3algebra

For $σ \in S_{n}$ , [F3] says that $T \mapsto σ T$ is a bijection of the tabloid set, so $β (σ T, σ U) = δ_{σ T, σ U} = δ_{T U} = β (T, U)$ on basis vectors, and bilinearity extends the identity to all $x, y$ .

1.3givenF2F4algebra

Every entry $β (e_{s}, e_{t})$ is a finite sum $\sum_{T} c_{s} (T) c_{t} (T)$ of products of tabloid coefficients, each of which lies in ${0, 1, - 1}$ by [F4]; hence $β (e_{s}, e_{t}) \in Z$ and, using the standard basis [F2], $G_{λ}$ is a matrix over $Z$ . Since $β$ is symmetric, so is $G_{λ}$ .

2.1givenF4step 1.2algebra

Applying step 1.2 to each term of $κ_{t}$ gives $β (κ_{t} x, y) = \sum_{γ \in C_{t}} sgn (γ) β (γ x, y) = \sum_{γ} sgn (γ) β (x, γ^{- 1} y)$ ; reindexing $γ \mapsto γ^{- 1}$ and using $sgn (γ^{- 1}) = sgn (γ)$ this is $β (x, κ_{t} y)$ , so $κ_{t}$ is self-adjoint for $β$ .

2.2givenF1F2step 1.3algebra

Let $R$ be a commutative ring. Under the identification of [F1], the $R$ -bilinear form $β_{R}$ with orthonormal tabloid basis sends $1 \otimes x, 1 \otimes y$ to $β (x, y)$ times $1_{R}$ , by expanding $x$ and $y$ in the tabloid basis; hence $β_{R}$ is the scalar extension of $β$ , and the matrix of $β_{R}$ in the standard basis $1 \otimes e_{s_{i}}$ is the scalar extension of $G_{λ}$ .

3.1givenF4F5step 1.3step 2.1step 2.2∎

Over the real field, $x = \sum_{T} a_{T} T$ with $a_{T} \in R$ satisfies $β (x, x) = \sum_{T} a_{T}^{2} \geq 0$ , with equality only when all $a_{T} = 0$ , i.e. $x = 0$ ; this is the positivity that holds on real vectors and it is the same quantity as the Hermitian form of [F5] on real coefficients, while the conjugate-linearity of [F5] and the positivity have no counterpart over a general commutative ring, and in particular cannot be reduced modulo a prime. The self-adjointness of step 2.1 and the integrality and scalar extension of steps 1.3 and 2.2 complete the asserted definition.

Depends on

Used by

Modular Specht modules need not be simple, and form heads can vanish Counterexample
Modular Specht form and radical quotient Definition
Decomposition matrices of S3 at p=2 and p=3 Example
Specht Gram rank for shape (2,2) Example
Conjugate Specht modules are sign-twisted duals over every field Lemma
Nonzero maps into tabloid quotients force dominance Lemma
Specht Gram gcd detects p-regularity Lemma
Dominance unitriangularity of the symmetric-group decomposition matrix Theorem
James submodule theorem over every field Theorem
Nonzero modular Specht quotient criterion Theorem

Dependency tree · two levels

19 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

Stacey Law, notes by Leonard Tomczak, Representation Theory of Symmetric Groups, §2.2 Lemma 2.3 (invariant symmetric bilinear tabloid form) and the Gram-matrix discussion, printed pp. 11-14 (standard reference, not scraped)
Charlotte Chan, Representation Theory of Symmetric Groups, Chapter 9, Definition 9.1 and Remark 9.2, printed pp. 31-32 (the bilinear version of the tabloid form) (standard reference, not scraped)