RemarkRemark: Literature-sourcedProof: 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.

Triangularity does not compute every modular decomposition number

Remark

The unitriangularity theorem of this page (Dominance unitriangularity of the symmetric-group decomposition matrix) is a constraint on the decomposition matrix, not a computation of it. It fixes three things: the positions that are forced to be zero, namely those with $d_{λ μ} = 0$ unless $μ ⊵ λ$ (Decomposition numbers and the decomposition matrix, Dominance order on partitions); the diagonal values $d_{λ λ} = 1$ for $p$ -regular $λ$ ; and the resulting lower unitriangular shape of the square block of $p$ -regular rows and columns. It says nothing about the value of an off-diagonal entry $d_{λ μ}$ at an allowed position, that is, at a pair with $μ ⊳ λ$ and $μ$ $p$ -regular: such an entry may be zero or positive, and the triangularity argument does not compute it.

This is a genuine limitation of the triangle data, not merely of the proof. For $n = 3$ the formal constraints are the same at $p = 2$ and at $p = 3$ : the $p$ -regular partitions used as column labels are $(3), (2, 1)$ in both characteristics, the forced zero at the position $((3), (2, 1))$ is present in both, and the same diagonal positions $d_{(3), (3)} = d_{(2, 1), (2, 1)} = 1$ are required. Yet the actual matrices, as recorded in James's Example 12.4, differ exactly at an allowed position, $d_{(2, 1), (3)} = 0$ for $p = 2$ and $d_{(2, 1), (3)} = 1$ for $p = 3$ . Hence a rule that reads off off-diagonal entries from the dominance-zero pattern and the diagonal alone cannot produce the decomposition matrix; the value depends on the modular composition factors of the Specht modules, which are separate input. The source separates the two problems in the same way: it opens the chapter on decomposition matrices by recording that there is "no known way of determining the composition factors of the general Specht module when the ground field $F$ has characteristic a prime $p$ ", and closes the paragraph with "The theorems we expound give only partial results" (James, §24, printed p. 98). This pair proves the triangularity constraints and a small number of explicit finite computations, and asserts no formula or algorithm for the general positive-characteristic decomposition matrix; the later Hecke-algebra and canonical-basis regimes involve further tools and hypotheses and are used nowhere on this pair.

Facts & Assumptions

Given: A prime $p$ , an integer $n \geq 0$ , and a splitting $p$ -modular system $(K, O, k)$ for $S_{n}$ (A splitting p-modular system for a finite group is a p-modular system whose fraction and residue fields split the needed group algebras). For the witness, $n = 3$ and $p = 2$ or $p = 3$ .

[F1]

By the unitriangularity theorem (Dominance unitriangularity of the symmetric-group decomposition matrix), with $d_{λ μ} = [S_{k}^{λ} : D^{μ}]$ the decomposition numbers of the ordinary irreducibles $S_{K}^{λ}$ (Decomposition numbers and the decomposition matrix): $d_{λ μ} = 0$ unless $μ ⊵ λ$ ; $d_{λ λ} = 1$ for every $p$ -regular $λ ⊢ n$ ; and with the $p$ -regular partitions listed in decreasing lexicographic order the leading square block is lower unitriangular.

[F2]

Combined with p-regular and p-restricted partitions: a partition is $p$ -regular when every positive part occurs fewer than $p$ times. For $n = 3$ one has $z_{1} (1, 1, 1) = 3 \geq p$ for $p \in {2, 3}$ , so $(1, 1, 1)$ is $p$ -singular for both primes; and $z_{3} (3) = 1 < p$ , $z_{2} (2, 1) = z_{1} (2, 1) = 1 < p$ for both, so $(3)$ and $(2, 1)$ are $p$ -regular for both primes. In the dominance order on partitions of $3$ one has $(3) ⊳ (2, 1) ⊳ (1, 1, 1)$ , and the dominance relation does not depend on $p$ (Dominance order on partitions).

[F3]

James's Example 12.4 (printed p. 43) records the decomposition matrices of $S_{3}$ with rows $S^{(3)}, S^{(2, 1)}, S^{(1, 1, 1)}$ and columns $D^{(3)}, D^{(2, 1)}$ ; at $p = 2$ $(\begin{array}{cc} 1 & 0 \\ 0 & 1 \\ 1 & 0 \end{array}), and at p = 3 (\begin{array}{cc} 1 & 0 \\ 1 & 1 \\ 0 & 1 \end{array}) .$ At $p = 2$ the equality $S_{k}^{(1, 1, 1)} ≅ S_{k}^{(3)}$ holds because the sign representation equals the trivial representation in characteristic $2$ ; at $p = 3$ the row $S^{(2, 1)}$ contains a trivial composition factor and a sign composition factor, each once.

[F4]

James §24 (printed p. 98) opens: "There is no known way of determining the composition factors of the general Specht module when the ground field $F$ has characteristic a prime $p$ ." The same paragraph closes: "The theorems we expound give only partial results."

Proof

technique · direct

1.1givenF1F2

By [F1] the triangle data for given $n$ and $p$ consist of: the index sets of rows and columns (all partitions of $n$ , and the $p$ -regular partitions of $n$ ), the set of positions forced to be zero ${(λ, μ) : μ ⋭ λ}$ , the diagonal positions ${(λ, λ) : λ p -regular}$ with value $1$ , and the lex ordering of the square block. No condition of [F1] assigns a value to an allowed position $(λ, μ)$ with $μ ⊳ λ$ , $μ$ $p$ -regular, $λ \neq μ$ ; in particular $μ ⋭ λ$ holds for $μ = (2, 1)$ , $λ = (3)$ in $n = 3$ , so the position $((3), (2, 1))$ is forced to be zero, while the position $((2, 1), (3))$ is allowed because $(3) ⊳ (2, 1)$ by [F2].

1.2givenF2F3

By [F2] the $p$ -regular partitions of $3$ are $(3), (2, 1)$ for $p = 2$ and for $p = 3$ , and $(1, 1, 1)$ is $p$ -singular for both primes; by [F3] the corresponding matrices $M_{p}$ with rows $(3), (2, 1), (1, 1, 1)$ and columns $(3), (2, 1)$ are $M_{2} = (\begin{array}{cc} 1 & 0 \\ 0 & 1 \\ 1 & 0 \end{array}), M_{3} = (\begin{array}{cc} 1 & 0 \\ 1 & 1 \\ 0 & 1 \end{array}) .$

2.1givenF1F2F3step 1.1step 1.2algebra

Both $M_{2}$ and $M_{3}$ satisfy every condition of step 1.1. Indeed, the left column is indexed by $(3)$ and the right column by $(2, 1)$ in both matrices; for the row $(3)$ the entry at column $(2, 1)$ is $0$ in both, as required since $(2, 1)$ does not dominate $(3)$ ; the diagonal entries $d_{(3), (3)} = 1$ and $d_{(2, 1), (2, 1)} = 1$ are present in both; and the row $(1, 1, 1)$ carries no diagonal requirement because $(1, 1, 1)$ is $p$ -singular for both primes, its entries lying in allowed positions. The two matrices nevertheless differ: $d_{(2, 1), (3)} = 0$ in $M_{2}$ and $d_{(2, 1), (3)} = 1$ in $M_{3}$ , at the position $((2, 1), (3))$ , which is allowed in both cases by step 1.1. Since the dominance pattern and the diagonal positions are the same while the entry differs, the conditions of [F1] do not determine the entries at allowed positions.

3.1givenF4step 2.1∎

By step 2.1 the triangularity theorem's data are strictly weaker than a determination of the decomposition matrix: they are satisfied by two different matrices, realized in characteristics $2$ and $3$ respectively, so an off-diagonal entry depends on the modular composition factors of the Specht modules and not on the triangular shape alone. A determination of the general off-diagonal entry therefore needs an additional input beyond the results of this pair, as the source's own separation of the two problems in [F4] records: the general composition factors are not determined by the theory expounded there, whose theorems "give only partial results". This pair establishes the unitriangularity constraints and the explicit small-shape computations only, and asserts no general formula or algorithm for decomposition numbers in positive characteristic; no step of its proofs computes an off-diagonal entry in the allowed region by a general rule.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

23 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, Example 12.4, printed p. 43 (decomposition matrices of S_3 at p=2 and p=3) (standard reference, not scraped)
G. D. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics 682, §24 opening, printed p. 98 (partial results and the general determination problem) (standard reference, not scraped)