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.

Murnaghan stability for Kronecker coefficients

Statement

Fix partitions $α, β, γ$ and work over a field $k$ of characteristic $0$ , with $V_{λ}$ denoting the irreducible $k S_{n}$ -module of shape $λ ⊢ n$ (Specht modules classify the irreducibles over every field of characteristic zero). Put $m (γ) : = \max {∣ α ∣ + α_{1}, ∣ β ∣ + β_{1}, ∣ γ ∣ + γ_{1}}$ and, for $n \geq m (γ)$ , let $g_{n} (α, β, γ) : = g_{α [n], β [n]}^{γ [n]}$ be the multiplicity of $V_{γ [n]}$ in $V_{α [n]} \otimes V_{β [n]}$ (Kronecker coefficients and the internal tensor product, Padded Kronecker coefficient sequences); set $g_{n} (α, β, γ) : = 0$ for $n < m (γ)$ , where the padded sequences are not partitions.

Then the sequence $n \mapsto g_{n} (α, β, γ)$ is eventually constant. More precisely, there is a finite set $S (α, β)$ of partitions and nonnegative integers ${\overset{ˉ}{g}}_{α, β}^{γ}$ for $γ \in S (α, β)$ , with ${\overset{ˉ}{g}}_{α, β}^{γ} : = 0$ for $γ \notin S (α, β)$ , such that $g_{n} (α, β, γ) = {\overset{ˉ}{g}}_{α, β}^{γ}$ for all sufficiently large $n$ and every partition $γ$ . No explicit bound on the stable rank is asserted here; the sharp global bound is a separate claim.

Facts & Assumptions

Given: Partitions $α, β, γ$ , a field $k$ of characteristic $0$ , and the FI-module machinery over $k$ .

[F1]

There is a finitely generated FI-module $V (α)$ over $k$ with $V (α)_{n} ≅ V_{α [n]}$ for $n \geq ∣ α ∣ + α_{1}$ and $V (α)_{n} = 0$ below; it has weight $∣ α ∣$ and stability degree $α_{1}$ (Each padded irreducible family is realised by a finitely generated FI-module).

[F2]

The pointwise tensor product of finitely generated FI-modules is finitely generated, with $(V \otimes W)_{S} = V_{S} \otimes_{k} W_{S}$ ; if $V$ is generated in degree $\leq m_{1}$ and $W$ in degree $\leq m_{2}$ , then $V \otimes W$ is generated in degree $\leq m_{1} + m_{2}$ (Tensor products of finitely generated FI-modules are finitely generated).

[F3]

A finitely generated FI-module over a characteristic-zero field with finite-dimensional values is uniformly representation stable, so the multiplicity of each admissible padded irreducible family is constant in a common stable range (Finite generation is equivalent to uniform representation stability, Uniform representation stability for consistent sequences of $S_{n}$ -modules).

[F4]

For partitions $λ \neq λ^{'}$ of possibly different sizes and $n \geq \max (∣ λ ∣ + λ_{1}, ∣ λ^{'} ∣ + λ_{1}^{'})$ , the irreducible modules $V_{λ [n]}$ and $V_{λ^{'} [n]}$ of $k S_{n}$ are nonisomorphic; and $V_{μ} \otimes V_{ν} ≅ ⨁_{γ} g_{μ, ν}^{γ} V_{γ}$ with $g_{μ, ν}^{γ}$ the Kronecker coefficient (Kronecker coefficients and the internal tensor product, First-row padding of a partition, Specht modules classify the irreducibles over every field of characteristic zero).

[F5]

Generation in a finite degree $d$ bounds the weight by $d$ : only tails $λ$ with $∣ λ ∣ \leq d$ can occur (Bounded generation degree bounds surjectivity degree and weight, Weight of an FI-module over a field of characteristic zero).

Proof

technique · direct

1.1F1

By [F1] the assignment $n \mapsto V_{α [n]}$ (and $0$ below the threshold) is a finitely generated FI-module $V (α)$ over $k$ with finite-dimensional values, and likewise $V (β)$ .

2.1F2F4step 1.1

By [F2] the pointwise tensor product $T : = V (α) \otimes V (β)$ is finitely generated and has finite-dimensional values; at every rank where both paddings are admissible its value is $V_{α [n]} \otimes_{k} V_{β [n]}$ . Its tensor multiplicities are the usual complex Kronecker coefficients: by the rational-model construction in Specht modules classify the irreducibles over every field of characteristic zero, $V_{μ} = k \otimes_{Q} S_{Q}^{μ}$ . The map $(c \otimes u) \otimes (e \otimes v) \mapsto c e \otimes (u \otimes v)$ identifies its tensor product with the scalar extension of the rational tensor product. Extend a rational Specht decomposition to $k$ and to $C$ ; tensoring carries a finite direct sum to the same sum of extended Specht modules (by the displayed map on bases), so the two fields have identical multiplicities.

3.1F3F5step 2.1

Since $T$ is finitely generated, it is generated in some finite degree $d$ . By [F5] only the finitely many partitions $λ$ with $∣ λ ∣ \leq d$ can label constituents. By [F3] their multiplicities are constant from some rank $N$ onward. Increase $N$ to make all these finitely many labels admissible. Then $T_{n} ≅ ⨁_{λ \in S} c_{λ} V_{λ [n]}$ for $n \geq N$ , where $S$ is the finite set of labels with positive multiplicity; put ${\overset{ˉ}{g}}_{α, β}^{γ} : = c_{γ}$ on $S$ and zero outside $S$ .

4.1F4step 3.1

Put $N_{0} : = \max {N, ∣ α ∣ + α_{1}, ∣ β ∣ + β_{1}, \max ({0} \cup {∣ λ ∣ + λ_{1} : λ \in S})}$ , finite because $S$ is finite. For $n \geq N_{0}$ the equality of step 3.1 holds with every $V_{γ [n]}$ , $γ \in S$ , an honest irreducible of $k S_{n}$ ; and the rank- $n$ value also decomposes by [F4] as $⨁_{γ} g_{α [n], β [n]}^{γ [n]} V_{γ [n]}$ , the sum over admissible tails $γ$ with $n \geq ∣ γ ∣ + γ_{1}$ , which label all partitions of $n$ uniquely by deleting their first row. By the nonisomorphism clause of [F4], the two decompositions have the same labels and multiplicities: $g_{n} (α, β, γ) = {\overset{ˉ}{g}}_{α, β}^{γ}$ for every partition $γ$ and every $n \geq N_{0}$ , where for $γ \notin S$ this says $g_{n} (α, β, γ) = 0$ .

5.1givenstep 4.1

The case $γ \notin S$ and $n < m (γ)$ also gives $g_{n} (α, β, γ) = 0$ by the convention in the statement, so making $N_{0}$ larger if necessary the equality $g_{n} (α, β, γ) = {\overset{ˉ}{g}}_{α, β}^{γ}$ holds for all $n \geq N_{0}$ and all partitions $γ$ ; this is eventual constancy with the finite support $S (α, β) : = S$ . The empty partition is included: $V_{\emptyset [n]}$ is the trivial module, the thresholds $∣ \emptyset ∣ + \emptyset_{1} = 0$ cause no special case, and the argument did not use any bound on $N_{0}$ sharper than existence.

6.1F1F2F3step 4.1∎

No choice principle is used: the FI-module suppliers are $Z$ -linear constructions on finite-dimensional representation categories, the finite family $S$ and its multiplicities come from [F3], and the comparison of irreducibles in step 4.1 is a finite-dimensional decomposition statement over $k$ ; the statement is asserted for every field of characteristic $0$ .

Depends on

Used by

The reduced (stable) Kronecker coefficient ḡ^γ_α,β Definition
Murnaghan stability for α=β=(2), ranks 4 to 9 Example
Brion's formula for the reduced Kronecker coefficients Theorem

Dependency tree · two levels

49 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.