TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pass

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.

The Murnaghan expansion at every admissible rank

Statement

For all partitions $α, β$ and every integer $n \geq \max {∣ α ∣ + α_{1}, ∣ β ∣ + β_{1}}$ : $s_{α [n]} * s_{β [n]} = \sum_{γ} {\overset{ˉ}{g}}_{α, β}^{γ} s_{γ [n]},$ an identity in the symmetric function ring, where $s_{γ [n]}$ is the Jacobi–Trudi determinant attached to the integer sequence $γ [n] = (n - ∣ γ ∣, γ_{1}, γ_{2}, \dots)$ , equal to $0$ or $\pm 1$ times a Schur function (The Jacobi–Trudi completion $s_{μ}$ , the operators $U_{n}$ and $V$ , and $stab (α, β)$ , First-row padding of a partition), and $*$ is the internal product (Kronecker coefficients and the internal tensor product).

For an ordinary partition $δ ⊢ n$ the coefficient of $s_{δ}$ is $g_{α [n], β [n]}^{δ} = \sum_{γ} {\overset{ˉ}{g}}_{α, β}^{γ} ⟨ s_{γ [n]}, s_{δ} ⟩_{H},$ where each Hall factor is $0$ or a sign coming from Jacobi–Trudi straightening (Schur functions form an orthonormal integral basis). At small admissible ranks the signed terms can cancel or coalesce; individual coefficients need not yet equal their reduced values. The range displayed is precisely the set of $n$ for which $α [n]$ and $β [n]$ are partitions.

Facts & Assumptions

Given: Partitions $α, β, γ, δ$ with $δ ⊢ n$ , and integers $k \geq 0$ .

[F1]

$R_{α, β} = \sum_{γ} {\overset{ˉ}{g}}_{α, β}^{γ} s_{γ}$ , a finite Schur expansion (Brion's formula for the reduced Kronecker coefficients, The reduced (stable) Kronecker coefficient ${\overset{ˉ}{g}}_{α, β}^{γ}$ ).

[F2]

$(U_{n} s_{α}) * (U_{n} s_{β}) = U_{n} R_{α, β}$ for every $n \geq 0$ , including nonadmissible ranks, and $\sum_{n \geq 0} (U_{n} s_{α}) * (U_{n} s_{β}) t^{n} = σ [t X] R_{α, β} [t X - 1]$ (The Jacobi–Trudi completion is compatible with the internal product).

[F3]

$U_{n} (s_{λ}) = s_{λ [n]}$ on the Schur basis and $U_{n}$ is $Z$ -linear; for $n \geq ∣ α ∣ + α_{1}$ the sequence $α [n]$ is a partition and $U_{n} s_{α} = s_{α [n]}$ is the honest Schur function (The Jacobi–Trudi completion $s_{μ}$ , the operators $U_{n}$ and $V$ , and $stab (α, β)$ , First-row padding of a partition).

[F4]

$s_{μ} * s_{ν} = \sum_{δ} g_{μ, ν}^{δ} s_{δ}$ for partitions of equal size, the Schur basis is orthonormal, and $s_{γ [n]}$ is an integer combination of Schur functions of degree $n$ that is $0$ or $\pm 1$ times a Schur function (Kronecker coefficients and the internal tensor product, Schur functions form an orthonormal integral basis, The Jacobi–Trudi completion $s_{μ}$ , the operators $U_{n}$ and $V$ , and $stab (α, β)$ ).

[F5]

The vertex-operator series identifies the coefficients of $σ [t X] R_{α, β} [t X - 1]$ , so $U_{n} R_{α, β}$ is the coefficient of $t^{n}$ there (Vertex-operator generating series for the padded Schur functions).

[F6]

The finite Littlewood expression for $R_{α, β}$ is given in The Jacobi–Trudi completion is compatible with the internal product; LR coefficients count lattice semistandard tableaux (The Littlewood–Richardson rule for products of Schur functions, Littlewood--Richardson tableaux and coefficients), and $s_{(m)}$ is the internal-product unit in degree $m$ (The internal (Kronecker) product on symmetric functions).

Proof

technique · direct

1.1F1

By [F1], $R_{α, β} = \sum_{γ} {\overset{ˉ}{g}}_{α, β}^{γ} s_{γ}$ with finitely many nonzero terms.

1.2F1F6

Local reduced coefficients for $(2), (2)$ . In the Littlewood expression of [F6], $(s_{δ} s_{τ})^{⊥} s_{(2)}$ is nonzero only when $δ = (m)$ and $τ = (r)$ are rows with $m + r \leq 2$ (a row LR target forces every factor to be a row); it then equals $s_{(2 - m - r)}$ . The analogous condition holds for $ϵ$ , and $s_{δ} * s_{ϵ} = 0$ unless their degrees agree, in which case it is $s_{(m)}$ by the unit rule. Here $(0)$ means $\emptyset$ . Hence $R_{(2), (2)} = \sum_{m + r \leq 2} s_{(2 - m - r)}^{2} s_{(m)} = s_{(2)}^{2} + s_{(1)}^{2} + 1 + s_{(1)}^{3} + s_{(1)} + s_{(2)}$ . Counting row LR tableaux gives $s_{(2)}^{2} = s_{(4)} + s_{(3, 1)} + s_{(2, 2)}$ and $s_{(1)}^{2} = s_{(2)} + s_{(1, 1)}$ ; adding one box to these two shapes gives $s_{(1)}^{3} = s_{(3)} + 2 s_{(2, 1)} + s_{(1, 1, 1)}$ . Thus $R_{(2), (2)} = 1 + s_{(1)} + 2 s_{(2)} + s_{(1, 1)} + s_{(1, 1, 1)} + 2 s_{(2, 1)} + s_{(2, 2)} + s_{(3)} + s_{(3, 1)} + s_{(4)}$ , whose coefficients are exactly the reduced coefficients by [F1].

2.1F2F3F5step 1.1

By [F2], $(U_{n} s_{α}) * (U_{n} s_{β}) = U_{n} R_{α, β}$ for every $n \geq 0$ ; by [F5], $U_{n} R_{α, β}$ is the coefficient of $t^{n}$ in $σ [t X] R_{α, β} [t X - 1]$ , and since $U_{n}$ is $Z$ -linear with $U_{n} (s_{γ}) = s_{γ [n]}$ by [F3], it equals $\sum_{γ} {\overset{ˉ}{g}}_{α, β}^{γ} s_{γ [n]}$ , a finite sum. Hence the displayed identity holds for every $n \geq 0$ at the level of Jacobi–Trudi completions.

3.1F3step 2.1

In the admissible range $n \geq \max {∣ α ∣ + α_{1}, ∣ β ∣ + β_{1}}$ the padded sequences $α [n], β [n]$ are partitions, so by [F3] the left side of step 2.1 is the internal product $s_{α [n]} * s_{β [n]}$ of two honest Schur functions; this proves the first display in exactly the stated range and shows that the range is the full set of $n$ for which both inputs are partitions.

3.2F3F4step 2.1step 1.2

Rank-four cancellation. Apply $U_{4}$ to the ten terms of step 1.2. The tails $\emptyset, (1), (2), (1, 1), (1, 1, 1)$ give $s_{(4)}, s_{(3, 1)}, s_{(2, 2)}, s_{(2, 1, 1)}, s_{(1, 1, 1, 1)}$ respectively. The tails $(2, 1), (2, 2)$ give zero by repeated straightening indices. The tails $(3), (3, 1), (4)$ give $- s_{(2, 2)}, - s_{(2, 1, 1)}, - s_{(3, 1)}$ respectively by [F3]–[F4]. With their multiplicities from step 1.2, the sum is $s_{(4)} + s_{(1, 1, 1, 1)} + s_{(2, 2)}$ ; in particular its $(2, 2)$ coefficient is $2 - 1 = 1$ , whereas the reduced $(2)$ coefficient is $2$ . This proves the announced cancellation locally.

4.1F4step 3.1

Coefficient form. For $δ ⊢ n$ , pair the identity of step 2.1 against $s_{δ}$ using orthonormality [F4]: the left side gives $⟨ s_{α [n]} * s_{β [n]}, s_{δ} ⟩_{H} = g_{α [n], β [n]}^{δ}$ by [F4], and the right side gives $\sum_{γ} {\overset{ˉ}{g}}_{α, β}^{γ} ⟨ s_{γ [n]}, s_{δ} ⟩_{H}$ , where each factor is a Hall pairing of the Jacobi–Trudi determinant $s_{γ [n]}$ with a Schur function, hence a signed multiplicity of a Schur constituent, $0$ or $\pm 1$ by [F4]. This is the second display, and it shows that individual coefficients can receive both positive and negative contributions from different $γ$ and need not equal ${\overset{ˉ}{g}}_{α, β}^{γ}$ .

5.1F1F2step 4.1∎

Choice and finiteness. Every step uses finite sums: the Schur expansion of $R_{α, β}$ is finite by [F1], the sum over $γ$ in step 2.1 is finite, and the coefficient extraction of step 4.1 uses the finite degree- $n$ part of the Schur basis. No form of the axiom of choice is used; the identity is asserted for all admissible ranks, not only for the eventually stable ones.

Remark

For $α = β = (2)$ , every rank $n \geq 4$ product is $U_{n} R_{(2), (2)}$ , where $R_{(2), (2)} = 1 + s_{(1)} + 2 s_{(2)} + s_{(1, 1)} + s_{(1, 1, 1)} + 2 s_{(2, 1)} + s_{(2, 2)} + s_{(3)} + s_{(3, 1)} + s_{(4)} .$ Its coefficients are the reduced coefficients, and all other reduced coefficients vanish. This finite polynomial and its rank-four cancellation are proved in steps 1.2 and 3.2 above.

Depends on

Used by

A global product threshold is not a per-coefficient sharpness claim Counterexample
Murnaghan stability for α=β=(2), ranks 4 to 9 Example
The exact global stabilization threshold for a fixed Kronecker product Theorem

Dependency tree · two levels

51 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

Emmanuel Briand, Rosa Orellana and Mercedes Rosas, The stability of the Kronecker product of Schur functions, arXiv:0907.4652v2 (2009); Murnaghan Theorem (2) p. 4, Example 1 p. 4 and Lemma 2.1 pp. 7-8 (standard reference, not scraped)
Emmanuel Briand, Amarpreeet Rattan and Mercedes Rosas, On the growth of the Kronecker coefficients, arXiv:1607.02887v1 (2016); section 2.3 pp. 5-6 (standard reference, not scraped)