LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

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.

The characteristic of a Young permutation character is complete homogeneous

Statement

For every $n \geq 0$ and $λ ⊢ n$ , let $φ^{λ}$ be the character of the Young permutation module $M^{λ}$ , the permutation module on $λ$ -tabloids, so that $M^{λ} ≅ {Ind}_{S_{λ}}^{S_{n}} 1$ (Young permutation modules are induced trivial modules, Young subgroups, tabloids, and permutation modules). Then

$ch (φ^{λ}) = h_{λ} \in Λ^{n},$

the product of the complete homogeneous symmetric functions of the parts of $λ$ .

Facts & Assumptions

Given: An integer $n \geq 0$ , a partition $λ = (λ_{1}, \dots, λ_{k}) ⊢ n$ with standard Young subgroup $S_{λ} \leq S_{n}$ , and a permutation $w \in S_{n}$ of cycle type $ρ ⊢ n$ .

[F1]

$M^{λ}$ is the permutation representation of $S_{n}$ on the finite set $Ω_{λ}$ of $λ$ -tabloids, and $M^{λ} ≅ {Ind}_{S_{λ}}^{S_{n}} 1$ as complex representations, so $φ^{λ}$ is the character of ${Ind}_{S_{λ}}^{S_{n}} 1$ (Young permutation modules are induced trivial modules).

[F2]

The character of a permutation representation on a finite $G$ -set $X$ is $χ_{C^{(X)}} (g) = # {x \in X : g \cdot x = x}$ (The character of a permutation representation counts fixed points).

[F3]

A $λ$ -tabloid is the row equivalence class ${t}$ of a $λ$ -tableau $t$ ; it records the unordered row sets, and $σ \cdot {t} : = {σ \cdot t}$ defines the left action of $S_{n}$ on $Ω_{λ}$ (Young subgroups, tabloids, and permutation modules).

[F4]

For partitions $λ, ρ$ of the same integer $n$ , the number $N (λ, ρ)$ counts the distributions of the cycles of a permutation of cycle type $ρ$ among the labelled rows with total row lengths $λ_{j}$ , cycles of equal length being distinct; and $h_{λ} = \sum_{ρ ⊢ n} N (λ, ρ) p_{ρ} / z_{ρ}$ in $Λ_{Q}$ , with $z_{ρ} = \prod_{i} i^{m_{i} (ρ)} m_{i} (ρ)!$ (Complete homogeneous functions expand in power sums with cycle-distribution coefficients).

[F5]

$h_{λ} = \prod_{j = 1}^{k} h_{λ_{j}}$ , with $h_{\emptyset} = 1$ ; the family ${h_{μ} : μ ⊢ n}$ is a $Z$ -basis of $Λ^{n}$ (Elementary and complete families freely generate the stable ring).

[F6]

For $f \in c f (S_{n})$ , $ch (f) = \sum_{ρ ⊢ n} f (ρ) p_{ρ} / z_{ρ}$ , where $f (ρ)$ is the common value of $f$ on elements of cycle type $ρ$ (The Frobenius characteristic map).

Proof

technique · direct

1.1F1F2given

By [F1] and [F2], $φ^{λ} (w)$ is the number of $λ$ -tabloids fixed by $w$ : $φ^{λ} (w) = # {{t} \in Ω_{λ} : w \cdot {t} = {t}}$ .

2.1F3step 1.1algebra

A $λ$ -tabloid ${t}$ with rows $B_{1}, \dots, B_{k}$ , where $B_{j}$ is the set of entries of the $j$ -th row of $t$ , is a partition of ${1, \dots, n}$ into labelled blocks with $∣ B_{j} ∣ = λ_{j}$ ; the tabloid is fixed by $w$ exactly when $w (B_{j}) = B_{j}$ for every $j$ , because equality of tabloids means equality of the row sets at each row index, even when rows have equal sizes. A subset of ${1, \dots, n}$ is $w$ -invariant if and only if it is a union of cycles of $w$ .

3.1F4step 2.1

It follows from step 2.1 that $φ^{λ} (w)$ is the number of ways to distribute the cycles of $w$ among the $k$ labelled rows with the $j$ -th row receiving cycles of total length $λ_{j}$ ; since the cycles of $w$ are distinct subsets of ${1, \dots, n}$ , cycles of equal length are distinct, and this number is exactly $N (λ, ρ)$ for $w$ of cycle type $ρ$ . Hence $φ^{λ} (ρ) = N (λ, ρ)$ for every $ρ ⊢ n$ .

4.1F4F5F6step 3.1algebra

Substituting $f = φ^{λ}$ into the definition of the characteristic and using step 3.1 and [F4], $ch (φ^{λ}) = \sum_{ρ ⊢ n} φ^{λ} (ρ) p_{ρ} / z_{ρ} = \sum_{ρ ⊢ n} N (λ, ρ) p_{ρ} / z_{ρ} = h_{λ}$ .

5.1F4F5F6step 4.1∎

For $n = 0$ one has $λ = \emptyset$ , $M^{\emptyset} = C$ is the trivial representation of the trivial group, $N (\emptyset, \emptyset) = 1$ , $z_{\emptyset} = 1$ and $h_{\emptyset} = 1$ , so $ch (φ^{\emptyset}) = p_{\emptyset} / z_{\emptyset} = 1 = h_{\emptyset}$ ; the identity of step 4.1 therefore holds for every $n \geq 0$ and every $λ ⊢ n$ .

Depends on

Used by

Outer induction is not the Kronecker product Counterexample
The Young permutation characteristic for shape (2,1) Example
The characteristic of a Specht character is a Schur function Theorem
The Frobenius characteristic is an isometric graded ring isomorphism Theorem

Dependency tree · two levels

26 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

I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Chapter I §7 (standard reference, not scraped)
G. D. James, The Representation Theory of the Symmetric Groups, §6 (standard reference, not scraped)