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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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First-row padding below the threshold is undefined

Statement refuted

Refuted claim: for every partition $α$ and every integer $n \geq ∣ α ∣$ the displayed sequence $(n - ∣ α ∣, α_{1}, α_{2}, \dots)$ is a partition of $n$ , so that the padded family and the padded Kronecker coefficient $g_{n} (α, β, γ)$ would be defined as soon as the size condition $n \geq ∣ α ∣$ holds (First-row padding of a partition).

The witness is $α = (2, 1)$ , of size $∣ α ∣ = 3$ and first part $α_{1} = 2$ . For $n = 3$ and $n = 4$ the size condition $n \geq ∣ α ∣$ holds, but the displayed sequences are $(3 - ∣ α ∣, 2, 1) = (0, 2, 1), (4 - ∣ α ∣, 2, 1) = (1, 2, 1),$ and neither is a partition: the first has a nonpositive leading entry, and the second is not weakly decreasing. Only from the threshold $∣ α ∣ + α_{1} = 3 + 2 = 5$ on does the family become a family of partitions, starting with $α [5] = (2, 2, 1)$ . Hence the padded sequence is undefined below the threshold, and for this shape the stable family begins at $n = 5$ .

Facts & Assumptions

Given: The partition $α = (2, 1)$ and the integers $n = 3, 4, 5$ (Partitions, English diagrams, and conjugation).

[F1]

Read $∣ \emptyset ∣ = \emptyset_{1} = 0$ . For a partition $α$ and an integer $n \geq ∣ α ∣ + α_{1}$ one sets $α [n] = (n - ∣ α ∣, α_{1}, α_{2}, \dots)$ , except that $\emptyset [0] = \emptyset$ ; this padding is a partition of $n$ . Below the threshold $α [n]$ is undefined (First-row padding of a partition).

[F2]

A partition of $n$ is a finite weakly decreasing sequence of positive integers with sum $n$ ; trailing zeros are not parts, so a displayed sequence that is not weakly decreasing, or that contains a nonpositive entry, is not a partition (Partitions, English diagrams, and conjugation).

[F3]

$α = (2, 1)$ has $∣ α ∣ = 2 + 1 = 3$ and $α_{1} = 2$ , so its padding threshold is $∣ α ∣ + α_{1} = 5$ . [F1, given]

[F4]

The size-indexed padded coefficient $g_{n} (α, β, γ) = g (α [n], β [n], γ [n])$ is defined exactly for $n \geq \max {∣ α ∣ + α_{1}, ∣ β ∣ + β_{1}, ∣ γ ∣ + γ_{1}}$ ; below this common threshold it is undefined (Padded Kronecker coefficient sequences).

Counterexample

technique · direct

1.1F2F3given

For $n = 3$ the displayed sequence is $(3 - 3, 2, 1) = (0, 2, 1)$ . Its leading entry $0$ is not a positive integer and $0 < 2$ shows that the sequence is not weakly decreasing, so by [F2] it is not a partition of $3$ , although $3 = ∣ α ∣$ and hence $n \geq ∣ α ∣$ holds.

1.2F2F3given

For $n = 4$ the displayed sequence is $(4 - 3, 2, 1) = (1, 2, 1)$ . Its entries are positive and sum to $4$ , but $1 < 2$ , so it is not weakly decreasing and by [F2] it is not a partition of $4$ , although $4 \geq ∣ α ∣$ holds.

2.1F1F2F3step 1.1step 1.2

For $n = 5$ , which is the threshold $∣ α ∣ + α_{1}$ computed in [F3], the displayed sequence is $(5 - 3, 2, 1) = (2, 2, 1)$ : the entries are positive, weakly decreasing and sum to $5$ , so it is a partition of $5$ . By [F1] the padding $α [5] = (2, 2, 1)$ is defined, and by step 1.1 and step 1.2 it is the first member of the family; the threshold is therefore exact.

3.1step 1.1step 1.2step 2.1F1F4∎

Consequently the refuted claim fails at $n = 3$ and $n = 4$ : the size condition $n \geq ∣ α ∣$ is not enough, and the padded family of $α = (2, 1)$ is a family of partitions only from $n = 5$ on. Any padded Kronecker coefficient $g_{n} (α, β, γ)$ with $α = (2, 1)$ and $n < 5$ is likewise undefined, because one of its three entries is not a partition.

Remarks

The failed threshold $∣ α ∣$ is exactly weaker than the true one $∣ α ∣ + α_{1}$ ; the extra term $α_{1}$ is what keeps the padded partition weakly decreasing, since the new first part $n - ∣ α ∣$ must be at least the old first part $α_{1}$ .
The example also shows why the size-indexed padded Kronecker coefficients of Padded Kronecker coefficient sequences are defined only from the maximum of the three thresholds: a single unpadded entry makes the whole expression undefined.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Thomas Church, Jordan S. Ellenberg and Benson Farb, FI-modules and stability for representations of symmetric groups, arXiv:1204.4533v4 (2014) (standard reference, not scraped)
Emmanuel Briand, Rosa Orellana and Mercedes Rosas, The stability of the Kronecker product of Schur functions, arXiv:0907.4652v2 (2009) (standard reference, not scraped)