CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passaudited 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.

p-regular and p-restricted labels differ

Statement refuted

The $p$ -regular and $p$ -restricted partitions of $n$ coincide, so that the James labelling $D^{λ}$ of the simple $k [S_{n}]$ -modules by $p$ -regular $λ$ and the labelling $D (μ)$ by $p$ -restricted $μ$ assign the same partition to each simple module, and a statement proved for one labelling applies verbatim to the other.

Facts & Assumptions

Given: The prime $p = 2$ and the partitions $(2)$ and $(1, 1)$ of $n = 2$ , with multiplicities $z_{j} (λ)$ of the positive parts and the conjugate partition $λ^{'}$ (p-regular and p-restricted partitions, Partitions, English diagrams, and conjugation).

[F1]

A partition $λ$ is $p$ -regular when $z_{j} (λ) < p$ for every $j \geq 1$ , and $p$ -restricted when $λ_{i} - λ_{i + 1} < p$ for every $i \geq 1$ , with the sequence padded by zeros; and $λ$ is $p$ -restricted if and only if $λ^{'}$ is $p$ -regular (p-regular and p-restricted partitions).

[F2]

The conjugate partition has parts $λ_{j}^{'} = # {i : λ_{i} \geq j}$ ; in particular the conjugate of a one-part partition is a column and conversely (Partitions, English diagrams, and conjugation).

[F3]

For a $p$ -regular partition $λ$ the James simple module $D^{λ}$ and the dual-label simple module $D (λ^{'})$ are related by $D^{λ} ≅ D (λ^{'}) \otimes sgn$ (p-regular and p-restricted labels under transpose and sign).

[F4]

The sign representation is the one-dimensional representation on which $σ$ acts by $sgn (σ) = \pm 1$ ; over a field of characteristic $2$ one has $- 1 = 1$ , so the sign representation is the trivial representation (The sign representation of $S_{n}$ and the restriction ${Res}_{H}^{G} (V)$ of a representation to a subgroup).

Counterexample

technique · direct

1.1givenF1algebra

For $λ = (2)$ one has $z_{2} (λ) = 1 < 2$ , so $(2)$ is $2$ -regular; and $λ_{1} - λ_{2} = 2 - 0 = 2$ , which is not $< 2$ , so $(2)$ is not $2$ -restricted. Thus $(2)$ is $2$ -regular but not $2$ -restricted.

1.2givenF1algebra

For $λ = (1, 1)$ one has $z_{1} (λ) = 2$ , which is not $< 2$ , so $(1, 1)$ is not $2$ -regular; and the padded differences are $λ_{1} - λ_{2} = 1 - 1 = 0 < 2$ and $λ_{2} - λ_{3} = 1 - 0 = 1 < 2$ , so $(1, 1)$ is $2$ -restricted. Thus $(1, 1)$ is $2$ -restricted but not $2$ -regular.

2.1givenF1F2step 1.1step 1.2algebra

By [F2], $(2)^{'} = (1, 1)$ and $(1, 1)^{'} = (2)$ : the diagram of $(2)$ has two columns of height $1$ , and the diagram of $(1, 1)$ has one column of height $2$ . This is exactly the conjugation exchange of [F1] that matches the two partitions of steps 1.1 and 1.2.

3.1givenF1F3F4step 1.1step 1.2step 2.1∎

Steps 1.1 and 1.2 exhibit partitions of the same integer $2$ that lie in exactly one of the two classes: $(2)$ is $2$ -regular and not $2$ -restricted, while $(1, 1)$ is $2$ -restricted and not $2$ -regular. Hence the two families do not coincide, and labelling by one of them is not labelling by the other. Moreover the two labels describe the same simple module: by [F3] applied to the $2$ -regular partition $(2)$ , whose conjugate $(1, 1)$ is $2$ -restricted, $D^{(2)} ≅ D ((1, 1)) \otimes sgn ≅ D ((1, 1)),$ the last step because the sign representation is trivial in characteristic $2$ by [F4]. So a statement about the $p$ -restricted label $D (μ)$ cannot be applied to the James label $D^{λ}$ without transposing the partition (and, in odd characteristic, inserting the sign twist); the change of partition is present already at $p = 2$ , where the sign twist itself is invisible.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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

G. D. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics 682, §10.1 and Lemma 10.2, printed pp. 36-37 (standard reference, not scraped)
Alexander Kleshchev, Representation Theory of Symmetric Groups and Related Hecke Algebras, Remark 5.5 (q=1 dictionary), PDF p. 25 (standard reference, not scraped)