CounterexampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)audited 2026-10-08

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The outer multiplicity is not the semistandard skew-tableau count

Statement

The claim refuted is that for all partitions $λ \supseteq μ$ and $ν$ with $∣ λ ∣ = ∣ μ ∣ + ∣ ν ∣$ , the multiplicity of $S^{λ}$ in the outer induction of $S^{μ} ⊠ S^{ν}$ equals the number of semistandard skew tableaux of shape $λ / μ$ and content $ν$ . This example has three such semistandard tableaux but outer multiplicity two.

Facts & Assumptions

Given: $λ = (4, 2, 1)$ , $μ = (2, 1)$ , and $ν = (3, 1)$ .

[F1]

The English diagram of a partition has row and column coordinates $(i, j)$ with rows numbered downward and columns rightward; $[λ]$ consists of $(i, j)$ with $1 \leq i$ and $1 \leq j \leq λ_{i}$ (Partitions, English diagrams, and conjugation).

[F2]

The skew diagram $[λ / μ]$ is $[λ] ∖ [μ]$ ; a semistandard skew tableau is weakly increasing along each row and strictly increasing down each column (Skew diagrams and semistandard skew tableaux).

[F3]

A tableau of content $ν = (3, 1)$ has exactly three entries equal to $1$ and one entry equal to $2$ (Semistandard tableaux and Kostka numbers).

[F4]

An LR tableau is semistandard and its reading word is read right-to-left in each row, from top to bottom; every prefix must have at least as many $i$ 's as $(i + 1)$ 's for each $i \geq 1$ . The LR coefficient counts these tableaux (Littlewood--Richardson tableaux and coefficients).

[F5]

The outer-induction multiplicity of $S^{λ}$ in the induced external product of $S^{μ}$ and $S^{ν}$ is exactly $c_{μ ν}^{λ}$ (The outer Littlewood–Richardson rule).

Counterexample

Counterexample technique: direct enumeration.

1.1F1F2F3

The diagrams give $[λ / μ] = {(1, 3), (1, 4), (2, 2), (3, 1)}$ , and the content condition [F3] requires three $1$ 's and one $2$ .

2.1F2F3step 1.1

A filling is determined by the position of its unique $2$ . Placing it at $(1, 3)$ is impossible: weak increase in the first row would force the entry at $(1, 4)$ to be at least $2$ , requiring a second $2$ . The other three assignments, listed as values in the box order $(1, 3), (1, 4), (2, 2), (3, 1)$ , are $(1, 2, 1, 1)$ , $(1, 1, 2, 1)$ , and $(1, 1, 1, 2)$ . Each is semistandard: the first row is weakly increasing and no two boxes lie in the same column. Thus there are exactly three semistandard fillings.

3.1F4step 2.1

Their reading words, in that order, are $2, 1, 1, 1$ , $1, 1, 2, 1$ , and $1, 1, 1, 2$ . The first fails the lattice condition at its first prefix; in each of the other two, every prefix has at least as many $1$ 's as $2$ 's. Since no entries exceed $2$ , the other lattice inequalities are automatic. Hence exactly two of the three semistandard fillings are LR tableaux by [F4].

4.1F4F5step 2.1step 3.1∎

Therefore $c_{(2, 1), (3, 1)}^{(4, 2, 1)} = 2$ by the LR-coefficient definition [F4], and the outer-induction multiplicity is also $2$ by [F5], while the semistandard skew-tableau count is $3$ by step 2.1. These unequal counts refute the stated claim.

Depends on

Used by

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Sources

I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford Mathematical Monographs, 1995 (standard reference, not scraped)
G. D. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics 682, Springer 1978 (standard reference, not scraped)