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Frobenius Characteristic and the Symmetric-Group Character Dictionary

1 · Prerequisites

2 · Summary

This page builds the classical dictionary between the ordinary character theory of the symmetric groups and the ring of symmetric functions. It begins with the graded abelian group $R_{S} = ⨁_{n} R (S_{n})$ of symmetric-group characters and the outer induction product $\circ$ , and it defines the Frobenius characteristic $ch (f) = \sum_{ρ} f (ρ) p_{ρ} / z_{ρ}$ on complex class functions, with the Hall form providing the metric on the symmetric function side. A local power-sum expansion of the complete homogeneous functions supplies the cycle-distribution coefficients used throughout.

The main structure theorem proves that $ch$ , restricted to the integral lattice $R_{S}$ , is an isometric isomorphism of graded rings from the outer-product ring onto $Λ$ : the isometry comes from the class sizes $n! / z_{ρ}$ , multiplicativity from Frobenius' induced-character formula and the split identity $z_{ρ} / (z_{μ} z_{ν}) = \prod_{i} (\binom{m_{i} (ρ)}{m_{i} (μ)})$ , and integrality and surjectivity from Young's rule together with the unitriangular Kostka change of basis and the integral basis of complete homogeneous functions. On Specht characters the dictionary reads $ch (χ^{λ}) = s_{λ}$ , so character values become power-sum coefficients, $χ^{λ} (ρ) = ⟨ s_{λ}, p_{ρ} ⟩_{H}$ ; the sign twist and the regular character appear as the omega involution and as $p_{1}^{n} = \sum_{λ} f^{λ} s_{λ}$ respectively.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

The graded ordinary representation ring of the symmetric groups

Definition

For $n \geq 0$ let $R (S_{n})$ be the character ring of the finite group $S_{n}$ , the $Z$ -span of its irreducible complex characters (Virtual characters and the character ring $R (G)$ of a finite group), so that $R (S_{0}) = Z \cdot 1$ for the trivial group $S_{0} = {1}$ (The finite symmetric group $S_{n}$ , one-line notation, and cycle notation). The graded ordinary representation ring of the symmetric groups is the direct sum

$R_{S} : = ⨁_{n \geq 0} R (S_{n}),$

the abelian group of finitely supported tuples $(f_{n})_{n \geq 0}$ with $f_{n} \in R (S_{n})$ and componentwise addition; an element $f \in R (S_{n})$ is homogeneous of degree $n$ , and the degree- $n$ component of an element $f = \sum_{n} f_{n}$ of $R_{S}$ is $f_{n}$ . This item defines only the graded abelian group and its degree decomposition: the multiplication used on $R_{S}$ is the outer induction product of The outer induction product of symmetric-group characters, not the tensor-product multiplication inside a single $R (S_{n})$ , and no ring axioms for the outer product are assumed here. We write

$R_{S, Q} : = Q \otimes_{Z} R_{S}, R_{S, C} : = C \otimes_{Z} R_{S}$

for the scalar extensions. No choice principle is used.

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

The outer induction product of symmetric-group characters

Definition

For $m, n \geq 0$ , $S_{m}$ is the symmetric group of ${1, \dots, m}$ , with $S_{0} = {1}$ (Young subgroups, tabloids, and permutation modules). Identify $S_{m} \times S_{n}$ (The external direct product $G \times H$ with componentwise multiplication) with the subgroup of $S_{m + n}$ preserving each of the two blocks: the first factor acts on ${1, \dots, m}$ and the second on ${m + 1, \dots, m + n}$ , so that $(σ, τ)$ acts as $σ$ on the first block and as $τ$ on the second after shifting its labels by $m$ . Either block may be empty; its symmetric group is trivial. These actions give an injective homomorphism, and every permutation preserving the blocks has a unique pair of restrictions, so its image is exactly the stated subgroup. For honest characters $χ$ of $S_{m}$ and $ψ$ of $S_{n}$ , let $χ ⊠ ψ$ be the character of $S_{m} \times S_{n}$ on the tensor product $V \otimes_{C} W$ of representations affording $χ$ and $ψ$ , with $(σ, τ) \cdot (v \otimes w) : = σ v \otimes τ w$ , so that

$(χ ⊠ ψ) (σ, τ) = χ (σ) ψ (τ)$

(The tensor product of two complex representations, Characters add on direct sums, multiply on tensor products, and conjugate on duals).

Every virtual character $f \in R (S_{m})$ is an integral combination $f = \sum_{i} a_{i} χ_{i}$ of the irreducible characters $χ_{i}$ of $S_{m}$ (Virtual characters and the character ring $R (G)$ of a finite group), and the coefficients $a_{i}$ are unique because the irreducible characters of a finite group are orthonormal, hence $Z$ -linearly independent, in $c f (S_{m})$ (The irreducible complex characters form an orthonormal basis of $c f (G)$ ). So for $f = \sum_{i} a_{i} χ_{i}$ and $g = \sum_{j} b_{j} ψ_{j}$ we may set

$f ⊠ g : = \sum_{i, j} a_{i} b_{j} (χ_{i} ⊠ ψ_{j}) \in R (S_{m} \times S_{n}),$

an integral combination of honest characters of $S_{m} \times S_{n}$ . Its value at $(σ, τ)$ is $f (σ) g (τ)$ by the displayed character formula, and uniqueness of the coefficients makes $f ⊠ g$ well defined. The assignment is $Z$ -bilinear in $(f, g)$ . The outer induction product is

$f \circ g : = \sum_{i, j} a_{i} b_{j} {Ind}_{S_{m} \times S_{n}}^{S_{m + n}} (χ_{i} ⊠ ψ_{j}) \in R (S_{m + n}),$

the induction of honest characters in the sense of The induced character ${Ind}_{H}^{G} χ$ of a complex character. The trivial character of $S_{0}$ is the unit of degree zero. This is the outer product, defined across different symmetric groups; the same-rank tensor product $f \cdot g$ on a single $R (S_{n})$ is a different operation, and the two are never conflated. No choice principle is used.

LemmaStatement: AI-adaptedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Complete homogeneous functions expand in power sums with cycle-distribution coefficients

Statement

Work in $Λ_{Q} = Q \otimes_{Z} Λ$ . For a partition $ρ$ write $m_{i} (ρ) : = # {j : ρ_{j} = i}$ and $z_{ρ} : = \prod_{i \geq 1} i^{m_{i} (ρ)} m_{i} (ρ)!$ (Partitions, English diagrams, and conjugation, Power sums $p_{k}$ and complete homogeneous symmetric polynomials $h_{k}$ ).

For partitions $λ$ and $ρ$ of the same integer, let $N (λ, ρ)$ be the number of ways to distribute the cycles of a permutation $w$ of cycle type $ρ$ among the $ℓ (λ)$ rows, labelled $1, \dots, ℓ (λ)$ , so that row $j$ receives cycles whose lengths sum to $λ_{j}$ . Cycles of equal length count as distinct here, because they are distinct cycles of $w$ ; equivalently,

$N (λ, ρ) = \sum_{(m_{i}^{(j)})} \prod_{i \geq 1} \frac{m_{i} (ρ)!}{\prod_{j} m_{i}^{(j)}!},$

the sum over all matrices $(m_{i}^{(j)})$ of nonnegative integers with $\sum_{j} m_{i}^{(j)} = m_{i} (ρ)$ for every $i$ and $\sum_{i} i m_{i}^{(j)} = λ_{j}$ for every $j$ ; each summand is the product over $i$ of the number of ways to assign the $m_{i} (ρ)$ labelled cycles of length $i$ to rows with the prescribed multiplicities $m_{i}^{(j)}$ , so $N (λ, ρ)$ is a nonnegative integer. Then

$h_{λ} = \sum_{ρ ⊢ ∣ λ ∣} N (λ, ρ) \frac{p_{ρ}}{z_{ρ}} in Λ_{Q} .$

In particular $h_{d} = \sum_{ρ ⊢ d} p_{ρ} / z_{ρ}$ for every $d \geq 0$ .

Facts & Assumptions

Given: Partitions $λ$ and $ρ$ of the same integer $n$ , with $λ = (λ_{1}, \dots, λ_{k})$ and $ρ ⊢ n$ .

[F1]

$Λ = ⨁_{d \geq 0} Λ^{d}$ , where an element of $Λ^{d}$ is a compatible sequence of degree- $d$ symmetric polynomials in $N$ variables whose transition maps set the last variables to zero, and multiplication is coordinatewise polynomial multiplication (The stable graded ring of symmetric functions).

[F2]

For every $d \geq 0$ the family ${h_{μ} : μ ⊢ d}$ is a $Z$ -basis of $Λ^{d}$ , where $h_{μ} : = \prod_{i} h_{μ_{i}}$ and $h_{\emptyset} = 1$ (Elementary and complete families freely generate the stable ring).

[F3]

For $k \geq 1$ the stable power sum $p_{k} \in Λ^{k}$ is the compatible sequence of the finite power sums $p_{k} (x_{1}, \dots, x_{N}) = x_{1}^{k} + \dots + x_{N}^{k}$ , and for a partition $μ$ one sets $p_{μ} : = \prod_{i} p_{μ_{i}}$ , with $p_{\emptyset} = 1$ ; likewise $h_{k} (x_{1}, \dots, x_{N}) = \sum_{a_{1} + \dots + a_{N} = k} x_{1}^{a_{1}} \dots x_{N}^{a_{N}}$ (Power sums $p_{k}$ and complete homogeneous symmetric polynomials $h_{k}$ ).

[F4]

For every $d \geq 0$ the family ${p_{μ} : μ ⊢ d}$ is a $Q$ -basis of $Λ_{Q}^{d} = Q \otimes_{Z} Λ^{d}$ (Power sums form a rational but not integral stable basis).

[F5]

A partition of $n$ is a weakly decreasing finite sequence of positive integers with sum $n$ ; the empty partition $\emptyset$ is the only partition of $0$ , and $m_{i} (μ)$ denotes the number of parts of $μ$ equal to $i$ (Partitions, English diagrams, and conjugation).

Proof

technique · direct

1.1F3algebra

Fix a rank $N \geq 0$ , work in $Q [x_{1}, \dots, x_{N}] [⁣ [t] ⁣]$ , and write $h_{d}^{(N)} : = h_{d} (x_{1}, \dots, x_{N})$ and $p_{k}^{(N)} : = p_{k} (x_{1}, \dots, x_{N})$ for the rank- $N$ specializations. The coefficient of $t^{d}$ in $\prod_{i = 1}^{N} (1 - x_{i} t)^{- 1}$ is $\sum_{a_{1} + \dots + a_{N} = d} x_{1}^{a_{1}} \dots x_{N}^{a_{N}} = h_{d}^{(N)}$ , so $\sum_{d \geq 0} h_{d}^{(N)} t^{d} = \prod_{i = 1}^{N} (1 - x_{i} t)^{- 1}$ ; taking the formal logarithm gives $\log \prod_{i = 1}^{N} (1 - x_{i} t)^{- 1} = \sum_{i = 1}^{N} \sum_{k \geq 1} x_{i}^{k} t^{k} / k = \sum_{k \geq 1} p_{k}^{(N)} t^{k} / k$ , and applying the formal exponential (with $\exp (\log U) = U$ for $U \in 1 + t Q [x_{1}, \dots, x_{N}] [⁣ [t] ⁣]$ and $\exp (A + B) = \exp (A) \exp (B)$ for series $A, B$ with zero constant term) yields $\prod_{i = 1}^{N} (1 - x_{i} t)^{- 1} = \exp (\sum_{k \geq 1} p_{k}^{(N)} t^{k} / k)$ . Expanding, $\exp (\sum_{k \geq 1} p_{k}^{(N)} t^{k} / k) = \prod_{k \geq 1} \exp (p_{k}^{(N)} t^{k} / k)$ , and in degree $t^{d}$ only the factors with $k \leq d$ contribute, so $h_{d}^{(N)}$ equals the sum of $\prod_{k \geq 1} (p_{k}^{(N)})^{m_{k}} / (k^{m_{k}} m_{k}!)$ over all tuples $(m_{k})_{k \geq 1}$ of nonnegative integers with $\sum_{k} k m_{k} = d$ ; grouping the tuple by the partition $ρ$ with $m_{k} (ρ) = m_{k}$ , and using $\prod_{k} p_{k}^{m_{k}} = p_{ρ}$ and $\prod_{k} k^{m_{k}} m_{k}! = z_{ρ}$ , gives $h_{d}^{(N)} = \sum_{ρ ⊢ d} p_{ρ}^{(N)} / z_{ρ}$ .

2.1F1F3step 1.1

For $M \geq N$ the transition map that sets $x_{N + 1}, \dots, x_{M}$ to zero sends $h_{d}^{(M)} \mapsto h_{d}^{(N)}$ and $p_{k}^{(M)} \mapsto p_{k}^{(N)}$ , by their finite-rank definitions; hence the rank- $N$ identities of step 1.1 are the projections of a single compatible sequence of degree- $d$ symmetric polynomials. Therefore $h_{d} = \sum_{ρ ⊢ d} p_{ρ} / z_{ρ}$ in $Λ_{Q}^{d}$ for every $d \geq 0$ , since two compatible sequences with equal projections are equal by [F1].

3.1F2step 2.1algebra

Let $λ = (λ_{1}, \dots, λ_{k}) ⊢ n$ . By definition $h_{λ} = \prod_{j = 1}^{k} h_{λ_{j}}$ [F2], and applying step 2.1 to each part gives $h_{λ_{j}} = \sum_{ρ^{(j)} ⊢ λ_{j}} p_{ρ^{(j)}} / z_{ρ^{(j)}}$ ; multiplying these $k$ finite sums, $h_{λ} = \sum_{(ρ^{(1)}, \dots, ρ^{(k)})} \prod_{j = 1}^{k} p_{ρ^{(j)}} / z_{ρ^{(j)}}$ , the sum over all $k$ -tuples of partitions with $ρ^{(j)} ⊢ λ_{j}$ .

4.1F4F5step 3.1algebra

The monomial $\prod_{j} p_{ρ^{(j)}}$ equals $p_{ρ}$ exactly when merging the parts of $ρ^{(1)}, \dots, ρ^{(k)}$ gives the multiset of parts of $ρ$ , that is, when $\sum_{j} m_{i}^{(j)} = m_{i} (ρ)$ for every $i$ , where $m_{i}^{(j)} : = m_{i} (ρ^{(j)})$ ; a $k$ -tuple is uniquely recovered from its matrix $(m_{i}^{(j)})$ by listing $m_{i}^{(j)}$ copies of each $i$ in decreasing order, and the further condition $\sum_{i} i m_{i}^{(j)} = λ_{j}$ records that $ρ^{(j)} ⊢ λ_{j}$ . Hence the coefficient of $p_{ρ}$ in $h_{λ}$ equals $\sum_{(m_{i}^{(j)})} \prod_{j} 1 / z_{ρ^{(j)}}$ , the sum over all matrices with those two properties; and for such a matrix $\sum_{j} m_{i}^{(j)} = m_{i} (ρ)$ gives $\prod_{j} i^{m_{i}^{(j)}} = i^{m_{i} (ρ)}$ , so $\prod_{j} 1 / z_{ρ^{(j)}} = (\prod_{i} m_{i} (ρ)! / \prod_{i, j} m_{i}^{(j)}!) / z_{ρ}$ . Therefore the coefficient of $p_{ρ}$ in $h_{λ}$ is $N (λ, ρ) / z_{ρ}$ , because the sum displayed in the statement counts, for each $i$ independently, the assignments of the $m_{i} (ρ)$ distinct cycles of length $i$ of a fixed permutation of cycle type $ρ$ to the $k$ labelled rows with the multiplicities $m_{i}^{(j)}$ . Since ${p_{ρ} : ρ ⊢ n}$ is a $Q$ -basis of $Λ_{Q}^{n}$ [F4], the coefficient comparison gives $h_{λ} = \sum_{ρ ⊢ n} N (λ, ρ) p_{ρ} / z_{ρ}$ in $Λ_{Q}^{n}$ .

5.1F2F3step 2.1step 4.1algebra∎

For $n = 0$ one has $λ = ρ = \emptyset$ , $k = 0$ , and the empty product conventions give $h_{\emptyset} = 1 = N (\emptyset, \emptyset) p_{\emptyset} / z_{\emptyset}$ , while for $λ = (d)$ the single row must receive every cycle, so $N ((d), ρ) = 1$ for every $ρ ⊢ d$ and step 4.1 gives $h_{d} = \sum_{ρ ⊢ d} p_{ρ} / z_{ρ}$ ; combined with step 2.1 this is the stated one-row case for every $d \geq 0$ , including $d = 0$ , where both sides equal $1$ . The general identity of the statement now follows from step 4.1 in all cases, with the empty partition handled by the computation just given.

DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-6.1-sol)Open item page →

The Frobenius characteristic map

Definition

Fix $n \geq 0$ and let $c f (S_{n})$ be the complex vector space of class functions on $S_{n}$ (Class functions and the complex vector space $c f (G)$ ). For $f \in c f (S_{n})$ and a partition $ρ ⊢ n$ let $f (ρ)$ denote the common value of $f$ on permutations of cycle type $ρ$ , which is well defined because cycle type determines the conjugacy class (The conjugacy classes of $S_{n}$ are indexed by the tuples $(c_{1}, \dots, c_{n})$ with $\sum k c_{k} = n$ ). Let

$z_{ρ} : = \prod_{i \geq 1} i^{m_{i} (ρ)} m_{i} (ρ)!,$

where $m_{i} (ρ)$ is the number of parts of $ρ$ equal to $i$ ; this is the order of the centralizer of an element of cycle type $ρ$ (If $σ \in S_{n}$ has $c_{k}$ cycles of length $k$ , then $∣ C_{S_{n}} (σ) ∣ = \prod_{k = 1}^{n} k^{c_{k}} c_{k}!$ ), so $z_{ρ}$ is a positive integer. The Frobenius characteristic of $f$ is

$ch (f) : = \sum_{ρ ⊢ n} f (ρ) \frac{p_{ρ}}{z_{ρ}} \in Λ_{C}^{n}, Λ_{C} : = C \otimes_{Z} Λ .$

The sum is finite and well defined because ${p_{ρ} : ρ ⊢ n}$ is a $Q$ -basis of $Λ_{Q}^{n}$ (Power sums form a rational but not integral stable basis); equivalently the family ${p_{ρ} / z_{ρ} : ρ ⊢ n}$ is a $Q$ -basis of $Λ_{Q}^{n}$ and is orthogonal for the Hall form, with $⟨ p_{ρ} / z_{ρ}, p_{σ} / z_{σ} ⟩_{H} = δ_{ρ σ} / z_{ρ}$ (Power sums are orthogonal for the Hall form). The codomain is $Λ_{C}$ , not $Λ_{Q}$ : for an arbitrary complex class function the coefficients $f (ρ) / z_{ρ}$ are complex. If all values of $f$ lie in $Q$ , then $ch (f) \in Λ_{Q}^{n}$ ; the dictionary theorems proved later on this page show that every virtual character of $S_{n}$ is rational-valued, and in fact integral-valued, so that its characteristic lies in the integral lattice $Λ^{n}$ .

Writing ${c f}_{S} : = ⨁_{n \geq 0} c f (S_{n})$ , the map $ch : {c f}_{S} \to Λ_{C}$ is defined degreewise by the displayed formula on each $c f (S_{n})$ . It is $C$ -linear on each summand, because evaluation $f \mapsto f (ρ)$ and scalar multiplication are linear. Its restriction to the character ring $R_{S} = ⨁_{n \geq 0} R (S_{n})$ is the Frobenius characteristic dictionary studied in the remaining items of this page. No choice principle is used.

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

The Frobenius characteristic is an isometry

Statement

Extend the Hall form on $Λ$ $Q$ -bilinearly to $Λ_{Q}$ and then sesquilinearly to $Λ_{C} = C \otimes_{Q} Λ_{Q}$ , linear in the first argument and conjugate-linear in the second, so that $⟨ p_{ρ}, p_{σ} ⟩_{H} = δ_{ρ σ} z_{ρ}$ for all partitions $ρ, σ$ (Power sums are orthogonal for the Hall form, The Hall inner product on symmetric functions). For all $f, g \in c f (S_{n})$ ,

$⟨ ch (f), ch (g) ⟩_{H} = ⟨ f, g ⟩_{S_{n}} = \frac{1}{n!} \sum_{w \in S_{n}} f (w) \overline{g (w)} .$

Consequently $ch$ is injective on $c f (S_{n})$ , and $⟨ f, f ⟩_{S_{n}} = \sum_{ρ ⊢ n} ∣ f (ρ) ∣^{2} / z_{ρ} \geq 0$ with equality if and only if $f = 0$ .

Facts & Assumptions

Given: An integer $n \geq 0$ and class functions $f, g \in c f (S_{n})$ .

[F1]

$ch (f) = \sum_{ρ ⊢ n} f (ρ) p_{ρ} / z_{ρ} \in Λ_{C}^{n}$ , where $f (ρ)$ is the common value of $f$ on elements of cycle type $ρ$ and $z_{ρ} = \prod_{i} i^{m_{i} (ρ)} m_{i} (ρ)!$ (The Frobenius characteristic map).

[F2]

A class function is constant on conjugacy classes, and a class function is determined by its values on one representative of each conjugacy class; the space $c f (S_{n})$ carries pointwise addition and scalar multiplication (Class functions and the complex vector space $c f (G)$ ).

[F3]

The standard inner product on $c f (S_{n})$ is $⟨ φ, ψ ⟩_{S_{n}} = \frac{1}{n!} \sum_{w \in S_{n}} φ (w) \overline{ψ (w)}$ , linear in the first argument and conjugate-linear in the second, and it is positive definite (The standard inner product on $c f (G)$ ).

[F4]

The Hall form is the graded $Z$ -bilinear form on $Λ$ with $⟨ h_{λ}, m_{μ} ⟩_{H} = δ_{λ μ}$ ; its $Q$ -bilinear extension to $Λ_{Q}$ satisfies $⟨ p_{ρ}, p_{σ} ⟩_{H} = δ_{ρ σ} z_{ρ}$ for all partitions $ρ, σ$ (The Hall inner product on symmetric functions, Power sums are orthogonal for the Hall form).

[F5]

If $σ \in S_{n}$ has exactly $m_{i} (ρ)$ cycles of length $i$ , then its centralizer has order $z_{ρ} = \prod_{i} i^{m_{i} (ρ)} m_{i} (ρ)!$ (If $σ \in S_{n}$ has $c_{k}$ cycles of length $k$ , then $∣ C_{S_{n}} (σ) ∣ = \prod_{k = 1}^{n} k^{c_{k}} c_{k}!$ ).

[F6]

The conjugacy classes of $S_{n}$ are indexed by the cycle types $ρ ⊢ n$ , and for $w \in S_{n}$ the class of $w$ has cardinality $[S_{n} : C_{S_{n}} (w)] = n! / ∣ C_{S_{n}} (w) ∣$ (The conjugacy classes of $S_{n}$ are indexed by the tuples $(c_{1}, \dots, c_{n})$ with $\sum k c_{k} = n$ , $G / C_{G} (x) \to {Cl}_{G} (x)$ is a bijection, so $∣ {Cl}_{G} (x) ∣ = [G : C_{G} (x)]$ whenever these cardinalities are finite).

Proof

technique · direct

1.1F4

The $Q$ -bilinear extension of the Hall form to $Λ_{Q}$ extends to a sesquilinear form on $Λ_{C} = C \otimes_{Q} Λ_{Q}$ by $⟨ a \otimes x, b \otimes y ⟩_{H} : = a \overline{b} ⟨ x, y ⟩_{H}$ on decomposable tensors; it is well defined because the form is $Q$ -bilinear, it is linear in the first argument and conjugate-linear in the second, and on power sums it has $⟨ p_{ρ} / z_{ρ}, p_{σ} / z_{σ} ⟩_{H} = δ_{ρ σ} / z_{ρ}$ by [F4] and $z_{ρ} > 0$ .

1.2F2F3F5F6algebra

On the group side, grouping the defining sum of [F3] by conjugacy classes, which by [F6] are indexed by the cycle types $ρ ⊢ n$ and have cardinality $n! / z_{ρ}$ by [F5] and [F6], and using that $f, g$ are constant on classes by [F2], gives $⟨ f, g ⟩_{S_{n}} = \frac{1}{n!} \sum_{ρ ⊢ n} \frac{n!}{z_{ρ}} f (ρ) \overline{g (ρ)} = \sum_{ρ ⊢ n} f (ρ) \overline{g (ρ)} / z_{ρ}$ .

2.1F1F4step 1.1algebra

Expanding both characteristics in the basis ${p_{ρ} / z_{ρ} : ρ ⊢ n}$ of $Λ_{Q}^{n}$ via [F1] and using the sesquilinearity of step 1.1 and the values $⟨ p_{ρ} / z_{ρ}, p_{σ} / z_{σ} ⟩_{H} = δ_{ρ σ} / z_{ρ}$ , $⟨ ch (f), ch (g) ⟩_{H} = \sum_{ρ, σ} f (ρ) \overline{g (σ)} ⟨ p_{ρ} / z_{ρ}, p_{σ} / z_{σ} ⟩_{H} = \sum_{ρ ⊢ n} f (ρ) \overline{g (ρ)} / z_{ρ}$ .

3.1F3step 2.1step 1.2∎

Steps 2.1 and 1.2 prove the displayed isometry $⟨ ch (f), ch (g) ⟩_{H} = ⟨ f, g ⟩_{S_{n}}$ for all $f, g \in c f (S_{n})$ . Taking $g = f$ and using positive definiteness of the standard inner product [F3], $⟨ f, f ⟩_{S_{n}} = \sum_{ρ ⊢ n} ∣ f (ρ) ∣^{2} / z_{ρ} \geq 0$ , with equality exactly when $f (ρ) = 0$ for every $ρ$ , that is, when $f = 0$ ; hence if $ch (f) = 0$ then $⟨ f, f ⟩_{S_{n}} = ⟨ ch (f), ch (f) ⟩_{H} = 0$ and $f = 0$ , so $ch$ is injective on $c f (S_{n})$ .

LemmaStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

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 $λ ⊢ n$ .

LemmaStatement: Literature-sourcedProof: AI-adaptedOpen item page →

The Frobenius characteristic preserves outer products

Statement

For all $m, n \geq 0$ and all $f \in R (S_{m})$ , $g \in R (S_{n})$ ,

$ch (f \circ g) = ch (f) ch (g) in Λ_{C}^{m + n},$

where $\circ$ is the outer induction product of The outer induction product of symmetric-group characters and the right-hand side is the algebra product in $Λ_{C}$ . If $f$ and $g$ are rational-valued, the identity holds in $Λ_{Q}^{m + n}$ .

Facts & Assumptions

Given: Integers $m, n \geq 0$ , honest characters $χ$ of $S_{m}$ and $ψ$ of $S_{n}$ , an element $w \in S_{m + n}$ of cycle type $ρ ⊢ m + n$ , and the block-preserving subgroup $H : = S_{m} \times S_{n} \leq S_{m + n}$ with blocks ${1, \dots, m}$ and ${m + 1, \dots, m + n}$ .

[F1]

For $f = \sum_{i} a_{i} χ_{i} \in R (S_{m})$ and $g = \sum_{j} b_{j} ψ_{j} \in R (S_{n})$ , the outer product is $f \circ g = \sum_{i, j} a_{i} b_{j} {Ind}_{S_{m} \times S_{n}}^{S_{m + n}} (χ_{i} ⊠ ψ_{j})$ , where $(χ_{i} ⊠ ψ_{j}) (σ, τ) = χ_{i} (σ) ψ_{j} (τ)$ ; the assignment $(f, g) \mapsto f \circ g$ is $Z$ -bilinear (The outer induction product of symmetric-group characters).

[F2]

The characteristic map is $ch (f) = \sum_{ρ} f (ρ) p_{ρ} / z_{ρ}$ , is defined on every class function and is linear, and $R (S_{m})$ is by definition the set of integral combinations of the honest (irreducible) characters of $S_{m}$ (The Frobenius characteristic map, Virtual characters and the character ring $R (G)$ of a finite group).

[F3]

Frobenius' formula: for a finite group $G$ , a subgroup $H \leq G$ and the character $θ$ of a finite-dimensional complex representation of $H$ , ${Ind}_{H}^{G} θ (g) = \frac{1}{∣ H ∣} \sum_{x \in G : x^{- 1} g x \in H} θ (x^{- 1} g x)$ for every $g \in G$ (Frobenius' formula for the character of an induced representation).

[F4]

For every $d \geq 0$ , ${p_{μ} : μ ⊢ d}$ is a $Q$ -basis of $Λ_{Q}^{d}$ , and $p_{μ} = \prod_{j} p_{μ_{j}}$ , with $p_{\emptyset} = 1$ (Power sums form a rational but not integral stable basis). Since each $z_{μ}$ is nonzero, ${p_{μ} / z_{μ} : μ ⊢ d}$ is also a rational basis; extending scalars to $C$ makes it a $C$ -basis of $Λ_{C}^{d}$ .

[F5]

Identify $S_{m} \times S_{n}$ with the subgroup of $S_{m + n}$ preserving the blocks ${1, \dots, m}$ and ${m + 1, \dots, m + n}$ . The two restrictions identify this subgroup with the direct product, including when a block is empty (The outer induction product of symmetric-group characters).

[F6]

For a partition $ρ$ , $z_{ρ} = \prod_{i} i^{m_{i} (ρ)} m_{i} (ρ)!$ is a positive integer (If $σ \in S_{n}$ has $c_{k}$ cycles of length $k$ , then $∣ C_{S_{n}} (σ) ∣ = \prod_{k = 1}^{n} k^{c_{k}} c_{k}!$ ).

Proof

technique · direct

1.1F1F2algebra

Both sides of the asserted identity are $Z$ -bilinear in $(f, g)$ : the outer product is bilinear by [F1], the characteristic map is linear by [F2], and multiplication in $Λ_{C}$ is bilinear. Since every element of $R (S_{m})$ is an integral combination of honest characters, and likewise for $R (S_{n})$ , it suffices to prove $ch (χ \circ ψ) = ch (χ) ch (ψ)$ for honest characters $χ$ of $S_{m}$ and $ψ$ of $S_{n}$ .

1.2F3F5

For honest $χ, ψ$ , the character $χ ⊠ ψ$ of $H$ is honest, so Frobenius' formula [F3] applied to $G = S_{m + n}$ and the subgroup $H$ of [F5] gives $(χ \circ ψ) (w) = \frac{1}{m! n!} \sum_{x \in S_{m + n} : x^{- 1} w x \in H} (χ ⊠ ψ) (x^{- 1} w x)$ .

2.1F5step 1.2algebra

The condition $x^{- 1} w x \in H$ says that $x^{- 1} w x$ preserves ${1, \dots, m}$ , equivalently that $w$ preserves $A : = x ({1, \dots, m})$ . Hence the $x$ occurring in the sum are exactly those with $x ({1, \dots, m}) = A$ for some $m$ -element $w$ -invariant subset $A \subseteq {1, \dots, m + n}$ , and for each such $A$ there are exactly $m! n!$ permutations $x$ with $x ({1, \dots, m}) = A$ . For all $x$ with the same $A$ , the permutation $x^{- 1} w x \in H$ has its $S_{m}$ -component conjugate through $x$ to $w ∣_{A}$ and its $S_{n}$ -component conjugate to $w ∣_{A^{c}}$ , so $(χ ⊠ ψ) (x^{- 1} w x) = χ (μ_{A}) ψ (ν_{A})$ , where $μ_{A}$ is the cycle type of $w ∣_{A}$ and $ν_{A}$ the cycle type of $w ∣_{A^{c}}$ ; this is independent of $x$ . With the factor $m! n! / m! n!$ cancelling, $(χ \circ ψ) (w) = \sum_{A} χ (μ_{A}) ψ (ν_{A})$ , the sum over the $m$ -element $w$ -invariant subsets $A$ .

3.1step 2.1algebra

An $m$ -element set $A$ is $w$ -invariant exactly when it is a union of cycles of $w$ , and then $∣ μ_{A} ∣ = m$ and the multisets of parts of $μ_{A}$ and $ν_{A}$ merge to the multiset of parts of $ρ$ . Conversely, every split $ρ = μ ⊎ ν$ with $∣ μ ∣ = m$ arises this way. For a fixed split, the number of $m$ -element $w$ -invariant $A$ with $μ_{A} = μ$ is $\prod_{i \geq 1} (\binom{m_{i} (ρ)}{m_{i} (μ)})$ , because for each cycle length $i$ one independently chooses which $m_{i} (μ)$ of the $m_{i} (ρ)$ cycles of $w$ of length $i$ are included in $A$ . Therefore $(χ \circ ψ) (ρ) = \sum_{μ ⊎ ν = ρ, ∣ μ ∣ = m} (\prod_{i \geq 1} (\binom{m_{i} (ρ)}{m_{i} (μ)})) χ (μ) ψ (ν)$ , an expression depending only on the cycle type $ρ$ of $w$ .

4.1F1F2F4F6step 3.1algebra

On the other side $ch (χ) ch (ψ) = \sum_{μ ⊢ m} \sum_{ν ⊢ n} χ (μ) ψ (ν) p_{μ} p_{ν} / (z_{μ} z_{ν})$ , using $p_{μ} p_{ν} = p_{μ ⊎ ν}$ from [F4], the coefficient of $p_{ρ} / z_{ρ}$ equals $\sum_{μ ⊎ ν = ρ} χ (μ) ψ (ν) z_{ρ} / (z_{μ} z_{ν})$ ; for a split $μ ⊎ ν = ρ$ one has $\frac{z_{ρ}}{z_{μ} z_{ν}} = \prod_{i \geq 1} \frac{m_{i} (ρ)!}{m_{i} (μ)! m_{i} (ν)!} = \prod_{i \geq 1} (\binom{m_{i} (ρ)}{m_{i} (μ)})$ , since $m_{i} (ν) = m_{i} (ρ) - m_{i} (μ)$ . This is exactly the coefficient in step 3.1; since ${p_{ρ} / z_{ρ} : ρ ⊢ m + n}$ is a $C$ -basis of $Λ_{C}^{m + n}$ by scalar extension [F4] and [F2] gives the same power-sum coefficients for the characteristic, $ch (χ \circ ψ) = ch (χ) ch (ψ)$ .

5.1F1F2step 1.1step 3.1step 4.1∎

By step 1.1 the identity holds for all virtual characters $f \in R (S_{m})$ and $g \in R (S_{n})$ . The cycle-split formula in step 3.1 also extends to these $f, g$ by bilinearity: $(f \circ g) (ρ) = \sum_{μ ⊎ ν = ρ, ∣ μ ∣ = m} (\prod_{i} (\binom{m_{i} (ρ)}{m_{i} (μ)})) f (μ) g (ν)$ . If $f, g$ are rational-valued, every term is rational, so $f \circ g$ is rational-valued. By [F2], $ch (f) \in Λ_{Q}^{m}$ and $ch (g) \in Λ_{Q}^{n}$ , while their product and $ch (f \circ g)$ lie in $Λ_{Q}^{m + n}$ . Thus the identity holds over $Q$ as claimed.

TheoremStatement: AI-adaptedProof: AI-adaptedOpen item page →

The Frobenius characteristic is an isometric graded ring isomorphism

Statement

Restrict the characteristic map to the integral lattice $R_{S} = ⨁_{n \geq 0} R (S_{n})$ of The graded ordinary representation ring of the symmetric groups. Then

$ch : R_{S} ⟶ Λ$

is a degree-preserving $Z$ -module isomorphism. It carries the outer induction product $\circ$ of The outer induction product of symmetric-group characters to multiplication, $ch (f \circ g) = ch (f) ch (g)$ , and the unit (the trivial character of $S_{0}$ ) to $1$ ; consequently $R_{S}$ is a commutative graded $Z$ -algebra and $ch$ is an isomorphism of graded rings onto $Λ$ . With the sesquilinear Hall form, $ch$ is an isometry as in The Frobenius characteristic is an isometry. After scalar extension, $ch \otimes Q : R_{S, Q} \to Λ_{Q}$ and $ch \otimes C : R_{S, C} \to Λ_{C}$ are isomorphisms. No choice principle is used.

Facts & Assumptions

Given: An integer $n \geq 0$ , the graded abelian group $R_{S} = ⨁_{n} R (S_{n})$ with the outer product $\circ$ , the characteristic map $ch$ on ${c f}_{S}$ , and the Specht characters $χ^{λ}$ of $S_{n}$ .

[F1]

$R (S_{n})$ is the character ring of $S_{n}$ , the integral span of its irreducible complex characters; $R_{S}$ is the direct sum of the $R (S_{n})$ with degree- $n$ homogeneous parts, and $R_{S, Q} = Q \otimes_{Z} R_{S}$ , $R_{S, C} = C \otimes_{Z} R_{S}$ (The graded ordinary representation ring of the symmetric groups).

[F2]

For $f = \sum_{i} a_{i} χ_{i} \in R (S_{m})$ , $g = \sum_{j} b_{j} ψ_{j} \in R (S_{n})$ , the outer product is $f \circ g = \sum_{i, j} a_{i} b_{j} {Ind}_{S_{m} \times S_{n}}^{S_{m + n}} (χ_{i} ⊠ ψ_{j})$ ; it is $Z$ -bilinear and maps $R (S_{m}) \times R (S_{n})$ into $R (S_{m + n})$ , and the trivial character of $S_{0}$ is the unit (The outer induction product of symmetric-group characters).

[F3]

$ch (f) = \sum_{ρ ⊢ n} f (ρ) p_{ρ} / z_{ρ}$ on $c f (S_{n})$ and $ch$ is linear on ${c f}_{S}$ ; the degree- $n$ component of $ch (f)$ for $f \in c f (S_{m})$ is zero when $n \neq m$ , so $ch$ preserves degrees (The Frobenius characteristic map).

[F4]

$⟨ ch (f), ch (g) ⟩_{H} = ⟨ f, g ⟩_{S_{n}}$ for all $f, g \in c f (S_{n})$ , $ch$ is injective on $c f (S_{n})$ , and $⟨ f, f ⟩_{S_{n}} = \sum_{ρ ⊢ n} ∣ f (ρ) ∣^{2} / z_{ρ}$ vanishes only for $f = 0$ (The Frobenius characteristic is an isometry).

[F5]

For every $μ ⊢ n$ , $ch (φ^{μ}) = h_{μ}$ , where $φ^{μ}$ is the character of the Young permutation module $M^{μ}$ (The characteristic of a Young permutation character is complete homogeneous).

[F6]

$ch (f \circ g) = ch (f) ch (g)$ for all $f \in R (S_{m})$ , $g \in R (S_{n})$ (The Frobenius characteristic preserves outer products).

[F7]

For every $d \geq 0$ , ${h_{μ} : μ ⊢ d}$ is a $Z$ -basis of $Λ^{d}$ (Elementary and complete families freely generate the stable ring).

[F8]

Young's rule: $M^{μ} ≅ ⨁_{λ ⊢ n} (S^{λ})^{\oplus K_{λ μ}}$ , so $φ^{μ} = \sum_{λ ⊢ n} K_{λ μ} χ^{λ}$ by additivity of characters (Young's rule for complex permutation modules, Characters add on direct sums, multiply on tensor products, and conjugate on duals).

[F9]

The matrix $(K_{λ μ})$ satisfies $h_{μ} = \sum_{λ} K_{λ μ} s_{λ}$ and is unitriangular in a linear extension of dominance, hence invertible over $Z$ (The Kostka change of basis is dominance-unitriangular).

[F10]

Every finite-dimensional complex representation of $S_{n}$ is completely reducible (Maschke's theorem over $C$ ), and the modules ${S^{λ} : λ ⊢ n}$ are pairwise inequivalent and exhaust the irreducible complex $S_{n}$ -representations; hence every honest character of $S_{n}$ is a nonnegative integral combination of the $χ^{λ}$ , and the family ${χ^{λ} : λ ⊢ n}$ is a $Z$ -basis of $R (S_{n})$ (Maschke's theorem for finite groups over fields whose characteristic does not divide $∣ G ∣$ , Specht modules classify the complex irreducibles of $S_{n}$ , Distinct complex Specht modules are inequivalent, Virtual characters and the character ring $R (G)$ of a finite group, Column antisymmetrizers, polytabloids, and Specht modules).

Proof

technique · direct

1.1F3F5F8F9F10

Membership $ch (R (S_{n})) \subseteq Λ^{n}$ : for $μ ⊢ n$ , [F8] gives $φ^{μ} = \sum_{λ} K_{λ μ} χ^{λ} \in R (S_{n})$ and [F5] gives $ch (φ^{μ}) = h_{μ} \in Λ^{n}$ . Since $(K_{λ μ})$ is invertible over $Z$ [F9], each $χ^{λ} = \sum_{μ} (K^{- 1})_{μ λ} φ^{μ}$ and, by linearity of $ch$ [F3], $ch (χ^{λ}) = \sum_{μ} (K^{- 1})_{μ λ} h_{μ} \in Λ^{n}$ . As the $χ^{λ}$ span $R (S_{n})$ over $Z$ [F10], $ch (R (S_{n})) \subseteq Λ^{n}$ for every $n$ , hence $ch (R_{S}) \subseteq Λ$ .

1.2F3F4

Injectivity: if $f = \sum_{n} f_{n} \in R_{S}$ satisfies $ch (f) = 0$ , then each degree component $ch (f_{n})$ vanishes by degree preservation [F3]; the isometry formula [F4] then gives $⟨ f_{n}, f_{n} ⟩_{S_{n}} = ⟨ ch (f_{n}), ch (f_{n}) ⟩_{H} = 0$ , and vanishing of $\sum_{ρ ⊢ n} ∣ f_{n} (ρ) ∣^{2} / z_{ρ}$ forces $f_{n} = 0$ ; hence $f = 0$ and $ch$ is injective on $R_{S}$ .

1.3F2F3F6

Multiplicativity and unit: for homogeneous $f \in R (S_{m})$ , $g \in R (S_{n})$ one has $ch (f \circ g) = ch (f) ch (g)$ [F6]; for general $f = \sum_{m} f_{m}$ , $g = \sum_{n} g_{n}$ bilinearity of $\circ$ [F2] and linearity of $ch$ [F3] give $ch (f \circ g) = \sum_{m, n} ch (f_{m}) ch (g_{n}) = ch (f) ch (g)$ . For the trivial character $e$ of $S_{0}$ one has $ch (e) = e (\emptyset) p_{\emptyset} / z_{\emptyset} = 1$ since $e (\emptyset) = 1$ and $z_{\emptyset} = p_{\emptyset} = 1$ .

2.1F5F7step 1.1

Surjectivity onto $Λ$ : step 1.1 shows $ch (R_{S}) \subseteq Λ$ , and $h_{μ} = ch (φ^{μ}) \in ch (R_{S})$ for every partition $μ$ [F5]; since ${h_{μ} : μ ⊢ d}$ is a $Z$ -basis of $Λ^{d}$ for every $d$ [F7], the image contains a $Z$ -basis of $Λ$ and therefore equals $Λ$ .

3.1F2F3F4step 1.2step 1.3step 2.1

By steps 1.1, 2.1 and 1.2 the map $ch : R_{S} \to Λ$ is a bijective degree-preserving $Z$ -linear map, hence a $Z$ -module isomorphism; by step 1.3 it is multiplicative and sends the unit to $1$ . Transporting the ring axioms of $Λ$ along the bijection: for $f, g, h \in R_{S}$ , associativity and commutativity of $\circ$ follow from $ch ((f \circ g) \circ h) = ch (f) ch (g) ch (h) = ch (f \circ (g \circ h))$ and $ch (f \circ g) = ch (g \circ f)$ together with injectivity of $ch$ , distributivity is the bilinearity of $\circ$ [F2], and $e \circ f = f$ because $ch (e \circ f) = 1 \cdot ch (f)$ ; so $R_{S}$ is a commutative graded $Z$ -algebra and $ch$ is a graded ring isomorphism. The isometry clause is [F4].

4.1F1step 3.1algebra∎

Both $R_{S}$ and $Λ$ are free $Z$ -modules, graded with finitely generated homogeneous components; a $Z$ -module isomorphism between them remains an isomorphism after tensoring with $Q$ or $C$ , with inverse ${ch}^{- 1} \otimes i d$ . Hence $ch \otimes Q : R_{S, Q} \to Λ_{Q}$ and $ch \otimes C : R_{S, C} \to Λ_{C}$ are isomorphisms.

TheoremStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

The characteristic of a Specht character is a Schur function

Statement

For every $n \geq 0$ and every $λ ⊢ n$ , let $S^{λ}$ be the complex Specht module of shape $λ$ (Column antisymmetrizers, polytabloids, and Specht modules) and $χ^{λ}$ its character, an irreducible character of $S_{n}$ (Specht modules classify the complex irreducibles of $S_{n}$ ). Then

$ch (χ^{λ}) = s_{λ} \in Λ^{n},$

the stable Schur function of shape $λ$ (Stable Schur functions from bialternants). In particular $ch$ maps the $Z$ -basis ${χ^{λ} : λ ⊢ n}$ of $R (S_{n})$ to the $Z$ -basis ${s_{λ} : λ ⊢ n}$ of $Λ^{n}$ , and all values $χ^{λ} (ρ)$ are integers.

Facts & Assumptions

Given: An integer $n \geq 0$ and partitions $λ, μ ⊢ n$ ; the Young permutation module $M^{μ}$ with character $φ^{μ}$ and the Specht modules $S^{λ}$ with characters $χ^{λ}$ .

[F1]

Young's rule: $M^{μ} ≅ ⨁_{λ ⊢ n} (S^{λ})^{\oplus K_{λ μ}}$ as $C S_{n}$ -modules, where $K_{λ μ}$ is the Kostka number (Young's rule for complex permutation modules).

[F2]

$K_{λ μ}$ is the number of semistandard $λ$ -tableaux of content $μ$ , so it is a nonnegative integer (Semistandard tableaux and Kostka numbers).

[F3]

Characters of finite-dimensional complex representations are additive on direct sums: $χ_{V \oplus W} = χ_{V} + χ_{W}$ (Characters add on direct sums, multiply on tensor products, and conjugate on duals).

[F4]

The characteristic map is $ch (f) = \sum_{ρ ⊢ n} f (ρ) p_{ρ} / z_{ρ}$ and is $Z$ -linear on class functions; $R (S_{n})$ is by definition the integral span of the irreducible characters of $S_{n}$ (The Frobenius characteristic map, Virtual characters and the character ring $R (G)$ of a finite group).

[F5]

$ch (φ^{μ}) = h_{μ}$ for every $μ ⊢ n$ (The characteristic of a Young permutation character is complete homogeneous).

[F6]

For partitions $λ, μ$ : $h_{μ} = \sum_{λ ⊢ n} K_{λ μ} s_{λ}$ , $K_{λ μ} = 0$ unless $λ ⊵ μ$ , and $K_{μ μ} = 1$ ; hence, in a linear extension of dominance from smaller to larger, $(K_{λ μ})$ is lower unitriangular with diagonal entries $1$ and is invertible over $Z$ (The Kostka change of basis is dominance-unitriangular).

[F7]

For every $d \geq 0$ the Schur functions ${s_{λ} : λ ⊢ d}$ form a $Z$ -basis of $Λ^{d}$ (Schur functions form an orthonormal integral basis).

[F8]

$h_{μ} = \sum_{ρ ⊢ n} N (μ, ρ) p_{ρ} / z_{ρ}$ in $Λ_{Q}^{n}$ , with $N (μ, ρ) \in Z$ and $z_{ρ} = \prod_{i} i^{m_{i} (ρ)} m_{i} (ρ)!$ (Complete homogeneous functions expand in power sums with cycle-distribution coefficients).

[F9]

The Specht modules $S^{λ}$ , $λ ⊢ n$ , are pairwise inequivalent and exhaust the irreducible complex representations of $S_{n}$ ; the irreducible complex characters of a finite group are orthonormal, hence $Z$ -linearly independent, in the space of class functions (Specht modules classify the complex irreducibles of $S_{n}$ , Distinct complex Specht modules are inequivalent, The irreducible complex characters form an orthonormal basis of $c f (G)$ ).

Proof

technique · direct

1.1F1F2F3

For every $μ ⊢ n$ , taking characters in Young's rule [F1] and using additivity on direct sums [F3] gives $φ^{μ} = \sum_{λ ⊢ n} K_{λ μ} χ^{λ}$ in $c f (S_{n})$ , the sum being finite.

2.1F4F5step 1.1algebra

Applying the $Z$ -linear map $ch$ to step 1.1 and using [F5] gives $h_{μ} = ch (φ^{μ}) = \sum_{λ ⊢ n} K_{λ μ} ch (χ^{λ})$ in $Λ_{C}^{n}$ .

3.1F6step 2.1algebra

Subtracting the identity $h_{μ} = \sum_{λ} K_{λ μ} s_{λ}$ of [F6] from step 2.1 yields $\sum_{λ ⊢ n} K_{λ μ} (ch (χ^{λ}) - s_{λ}) = 0$ for every $μ ⊢ n$ , a homogeneous linear system with coefficient matrix $K^{T}$ , where $K = (K_{λ μ})$ ; since $K$ is unitriangular in a linear extension of dominance, both $K$ and $K^{T}$ are invertible over $Z$ , so the only solution is the zero vector and $ch (χ^{λ}) = s_{λ}$ for every $λ ⊢ n$ .

4.1F4F8step 3.1algebra

Inverting the integral matrices, $s_{λ} = \sum_{μ ⊢ n} (K^{- 1})_{μ λ} h_{μ}$ with $(K^{- 1})_{μ λ} \in Z$ ; substituting $h_{μ} = \sum_{ρ ⊢ n} N (μ, ρ) p_{ρ} / z_{ρ}$ with integral $N (μ, ρ)$ gives $s_{λ} = \sum_{ρ ⊢ n} c_{λ ρ} p_{ρ} / z_{ρ}$ with $c_{λ ρ} = \sum_{μ} (K^{- 1})_{μ λ} N (μ, ρ) \in Z$ . Since ${p_{ρ} / z_{ρ} : ρ ⊢ n}$ is a $Q$ -basis of $Λ_{Q}^{n}$ and $ch (χ^{λ}) = \sum_{ρ} χ^{λ} (ρ) p_{ρ} / z_{ρ}$ by definition, comparing coefficients gives $χ^{λ} (ρ) = c_{λ ρ} \in Z$ for every $ρ ⊢ n$ .

5.1F4F7F9step 3.1step 4.1∎

The characters $χ^{λ}$ are pairwise distinct irreducible characters of $S_{n}$ and the irreducible characters are $Z$ -linearly independent [F9]; since $R (S_{n})$ is by definition their integral span [F4], the family ${χ^{λ} : λ ⊢ n}$ is a $Z$ -basis of $R (S_{n})$ . By [F7] the family ${s_{λ} : λ ⊢ n}$ is a $Z$ -basis of $Λ^{n}$ , and by step 3.1 the map $ch$ carries the first basis bijectively onto the second; step 4.1 shows that all character values are integers.

CorollaryStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Irreducible symmetric-group character values are power-sum coefficients

Statement

For all $λ, ρ ⊢ n$ ,

$χ^{λ} (ρ) = ⟨ s_{λ}, p_{ρ} ⟩_{H},$

the coefficient of $p_{ρ} / z_{ρ}$ in the power-sum expansion of $s_{λ}$ . Equivalently,

$s_{λ} = \sum_{ρ ⊢ n} χ^{λ} (ρ) \frac{p_{ρ}}{z_{ρ}} .$

Facts & Assumptions

Given: An integer $n \geq 0$ and partitions $λ, ρ ⊢ n$ .

[F1]

$ch (χ^{λ}) = s_{λ}$ in $Λ^{n}$ , where $χ^{λ}$ is the character of the Specht module $S^{λ}$ and all its values are integers (The characteristic of a Specht character is a Schur function).

[F2]

$ch (f) = \sum_{σ ⊢ n} f (σ) p_{σ} / z_{σ}$ for $f \in c f (S_{n})$ , where $z_{σ} = \prod_{i} i^{m_{i} (σ)} m_{i} (σ)! > 0$ (The Frobenius characteristic map).

[F3]

The Hall form is the $Z$ -bilinear form with $⟨ h_{α}, m_{β} ⟩_{H} = δ_{α β}$ and $Q$ -bilinear extension to $Λ_{Q}$ , and $⟨ p_{σ}, p_{ρ} ⟩_{H} = δ_{σ ρ} z_{ρ}$ (The Hall inner product on symmetric functions, Power sums are orthogonal for the Hall form).

Proof

technique · direct

1.1F1F2

By [F1] and the definition of the characteristic [F2], $s_{λ} = ch (χ^{λ}) = \sum_{σ ⊢ n} χ^{λ} (σ) p_{σ} / z_{σ}$ in $Λ_{Q}^{n}$ .

2.1F3step 1.1algebra

Pairing both sides of step 1.1 with $p_{ρ}$ and using $Q$ -bilinearity of the Hall form and the orthogonality $⟨ p_{σ}, p_{ρ} ⟩_{H} = δ_{σ ρ} z_{ρ}$ of [F3], $⟨ s_{λ}, p_{ρ} ⟩_{H} = \sum_{σ ⊢ n} \frac{χ^{λ} (σ)}{z_{σ}} ⟨ p_{σ}, p_{ρ} ⟩_{H} = \frac{χ^{λ} (ρ)}{z_{ρ}} z_{ρ} = χ^{λ} (ρ)$ .

3.1step 1.1step 2.1∎

Step 2.1 is the first displayed identity; step 1.1 exhibits $χ^{λ} (ρ)$ as the coefficient of $p_{ρ} / z_{ρ}$ in the expansion of $s_{λ}$ in the basis ${p_{σ} / z_{σ} : σ ⊢ n}$ of $Λ_{Q}^{n}$ , which is the equivalent second display.

PropositionStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

Sign twist corresponds to the omega involution

Statement

Let $sgn$ be the sign representation of $S_{n}$ and let $V$ be a finite-dimensional complex representation of $S_{n}$ with character $χ_{V}$ and characteristic $F = ch (χ_{V})$ . Then

$ch (χ_{V \otimes sgn}) = ω (F),$

where $ω$ is the involutive algebra endomorphism of $Λ$ with $ω (e_{r}) = h_{r}$ , extended to $Λ_{C}$ by complex linearity. In particular, for every $λ ⊢ n$ ,

$ch (χ_{S^{λ} \otimes sgn}) = ω (s_{λ}) = s_{λ^{'}},$

so $S^{λ} \otimes sgn ≅ S^{λ^{'}}$ and $χ^{λ^{'}} (w) = (- 1)^{n - ℓ (ρ)} χ^{λ} (w)$ on elements of cycle type $ρ$ .

Facts & Assumptions

Given: An integer $n \geq 0$ , a finite-dimensional complex representation $V$ of $S_{n}$ with character $χ_{V}$ , the sign representation $sgn$ , and $w \in S_{n}$ of cycle type $ρ ⊢ n$ with $ℓ (ρ)$ parts.

[F1]

$ch (χ) = \sum_{ρ ⊢ n} χ (ρ) p_{ρ} / z_{ρ}$ for a character $χ$ , where $χ (ρ)$ is its value on cycle type $ρ$ and $z_{ρ} = \prod_{i} i^{m_{i} (ρ)} m_{i} (ρ)!$ (The Frobenius characteristic map).

[F2]

For finite-dimensional complex representations $V, W$ of $S_{n}$ one has $χ_{V \otimes W} (g) = χ_{V} (g) χ_{W} (g)$ for every $g$ , where $V \otimes W$ is the tensor product representation with $g \cdot (v \otimes w) = g v \otimes g w$ (Characters add on direct sums, multiply on tensor products, and conjugate on duals, The tensor product of two complex representations).

[F3]

The sign representation of $S_{n}$ is one-dimensional with $σ \cdot a = sgn (σ) a$ ; a $k$ -cycle has sign $(- 1)^{k - 1}$ , and $sgn (w) = (- 1)^{n - c (w)}$ , where $c (w) = ℓ (ρ)$ is the number of cycles of $w$ counted with fixed points (The sign representation of $S_{n}$ and the restriction ${Res}_{H}^{G} (V)$ of a representation to a subgroup, A $k$ -cycle has sign $(- 1)^{k - 1}$ , and $sgn (σ) = (- 1)^{n - c (σ)}$ when fixed points are counted as cycles).

[F4]

The $Z$ -algebra endomorphism $ω : Λ \to Λ$ with $ω (e_{r}) = h_{r}$ is an involution and satisfies $ω (p_{r}) = (- 1)^{r - 1} p_{r}$ in $Λ_{Q}$ and $ω (s_{λ}) = s_{λ^{'}}$ for every partition $λ$ ; it extends to a $C$ -linear algebra endomorphism of $Λ_{C}$ (The omega involution conjugates Schur functions).

[F5]

$ch (χ^{λ}) = s_{λ}$ for every $λ ⊢ n$ , where $χ^{λ}$ is the character of the Specht module $S^{λ}$ (The characteristic of a Specht character is a Schur function).

[F6]

Two finite-dimensional complex representations of a finite group are isomorphic if and only if their characters are equal (Finite-dimensional complex representations of a finite group are determined up to isomorphism by their characters).

[F7]

The characteristic map is injective on $c f (S_{n})$ (The Frobenius characteristic is an isometry).

Proof

technique · direct

1.1F2F3

By [F2] and [F3], for $w$ of cycle type $ρ$ the tensor-product character is $χ_{V \otimes sgn} (w) = χ_{V} (w) sgn (w) = χ_{V} (ρ) (- 1)^{n - ℓ (ρ)}$ .

1.2F4given

Since $ω$ is a $C$ -algebra endomorphism of $Λ_{C}$ [F4] and $p_{ρ} = \prod_{i} p_{ρ_{i}}$ with $ℓ (ρ)$ factors, $ω (p_{ρ}) = \prod_{i} ω (p_{ρ_{i}}) = \prod_{i} (- 1)^{ρ_{i} - 1} p_{ρ_{i}} = (- 1)^{\sum_{i} (ρ_{i} - 1)} p_{ρ} = (- 1)^{n - ℓ (ρ)} p_{ρ}$ .

2.1F1step 1.1

From the definition of the characteristic [F1] and step 1.1, $ch (χ_{V \otimes sgn}) = \sum_{ρ ⊢ n} χ_{V \otimes sgn} (ρ) p_{ρ} / z_{ρ} = \sum_{ρ ⊢ n} χ_{V} (ρ) (- 1)^{n - ℓ (ρ)} p_{ρ} / z_{ρ}$ .

2.2F1F4step 1.2

By $C$ -linearity of $ω$ [F4], [F1] and step 1.2, $ω (F) = ω (\sum_{ρ ⊢ n} χ_{V} (ρ) p_{ρ} / z_{ρ}) = \sum_{ρ ⊢ n} χ_{V} (ρ) ω (p_{ρ}) / z_{ρ} = \sum_{ρ ⊢ n} χ_{V} (ρ) (- 1)^{n - ℓ (ρ)} p_{ρ} / z_{ρ}$ .

3.1step 2.1step 2.2

Steps 2.1 and 2.2 exhibit two equal expressions, so $ch (χ_{V \otimes sgn}) = ω (F)$ for every finite-dimensional complex representation $V$ of $S_{n}$ .

4.1F4F5step 3.1

Taking $V = S^{λ}$ in step 3.1 and using [F5], $ch (χ_{S^{λ} \otimes sgn}) = ω (s_{λ}) = s_{λ^{'}}$ by [F4].

5.1F5F6F7step 1.1step 4.1∎

Since $ch$ is injective on $c f (S_{n})$ [F7] and $ch (χ^{λ^{'}}) = s_{λ^{'}}$ [F5], step 4.1 gives $χ_{S^{λ} \otimes sgn} = χ^{λ^{'}}$ ; by [F6] this equality of characters is equivalent to $S^{λ} \otimes sgn ≅ S^{λ^{'}}$ . Evaluating the character identity from step 1.1 for $V = S^{λ}$ gives $χ^{λ^{'}} (w) = (- 1)^{n - ℓ (ρ)} χ^{λ} (w)$ for $w$ of cycle type $ρ$ .

PropositionStatement: Literature-sourcedProof: AI-adaptedjudge pass (gpt-6.1-sol)Open item page →

The regular character has characteristic $p_{1}^{n}$

Statement

Let $χ_{r e g}$ be the regular character of $S_{n}$ and let $f^{λ} : = \dim_{C} S^{λ}$ , the number of standard $λ$ -tableaux (Standard polytabloids form a basis of a complex Specht module). Then

$ch (χ_{r e g}) = p_{1}^{n}, and expanding in the Schur basis p_{1}^{n} = \sum_{λ ⊢ n} f^{λ} s_{λ} .$

Facts & Assumptions

Given: An integer $n \geq 0$ , the regular representation $C [S_{n}]$ with character $χ_{r e g}$ , and the Specht modules $S^{λ}$ with characters $χ^{λ}$ and dimensions $f^{λ} = \dim_{C} S^{λ}$ .

[F1]

$ch (f) = \sum_{ρ ⊢ n} f (ρ) p_{ρ} / z_{ρ}$ for class functions $f \in c f (S_{n})$ , with $z_{ρ} = \prod_{i} i^{m_{i} (ρ)} m_{i} (ρ)!$ positive (The Frobenius characteristic map).

[F2]

$χ_{r e g} (1) = n! = ∣ S_{n} ∣$ and $χ_{r e g} (w) = 0$ for $w \neq 1$ (The regular character is $∣ G ∣$ at $1$ and $0$ away from $1$ ).

[F3]

$C [S_{n}]$ is a finite-dimensional complex representation of the finite group $S_{n}$ and is completely reducible, by Maschke's theorem applied to the characteristic-zero field $C$ (Maschke's theorem for finite groups over fields whose characteristic does not divide $∣ G ∣$ ).

[F4]

The modules ${S^{λ} : λ ⊢ n}$ form a complete irredundant list, up to isomorphism, of the irreducible complex $S_{n}$ -representations (Specht modules classify the complex irreducibles of $S_{n}$ , Distinct complex Specht modules are inequivalent).

[F5]

If $V ≅ ⨁_{j} m_{j} V_{j}$ with $V_{j}$ a complete set of representatives of the irreducible representations and $χ_{j} = χ_{V_{j}}$ , then $m_{j} = ⟨ χ_{V}, χ_{j} ⟩$ (The multiplicity of an irreducible summand is a character inner product).

[F6]

The standard inner product is $⟨ φ, ψ ⟩ = \frac{1}{n!} \sum_{w \in S_{n}} φ (w) \overline{ψ (w)}$ (The standard inner product on $c f (G)$ ).

[F7]

$\dim_{C} S^{λ} = f^{λ}$ , the number of standard $λ$ -tableaux (Standard polytabloids form a basis of a complex Specht module).

[F8]

$ch (χ^{λ}) = s_{λ}$ for every $λ ⊢ n$ (The characteristic of a Specht character is a Schur function).

Proof

technique · direct

1.1F2given

By [F2], $χ_{r e g}$ vanishes except at the identity, whose cycle type is $(1^{n})$ ; for that partition $m_{1} ((1^{n})) = n$ , so $z_{(1^{n})} = 1^{n} \cdot n! = n!$ .

1.2F3F4F5F6

By [F3] the regular representation is completely reducible, and by [F4] its irreducible summands are copies of the Specht modules, so $C [S_{n}] ≅ ⨁_{λ ⊢ n} m_{λ} S^{λ}$ for nonnegative integers $m_{λ}$ ; by [F5] and [F6] each multiplicity is $m_{λ} = ⟨ χ_{r e g}, χ^{λ} ⟩ = \frac{1}{n!} \sum_{w \in S_{n}} χ_{r e g} (w) \overline{χ^{λ} (w)}$ , and [F2] reduces the sum to its identity term.

2.1F1step 1.1algebra

Substituting $f = χ_{r e g}$ into [F1] and using step 1.1, $ch (χ_{r e g}) = \sum_{ρ ⊢ n} χ_{r e g} (ρ) p_{ρ} / z_{ρ} = n! p_{(1^{n})} / n! = p_{1}^{n}$ , since $p_{(1^{n})} = p_{1}^{n}$ by the product convention for power sums.

2.2F5F6F7step 1.2algebra

By step 1.2, $m_{λ} = \frac{1}{n!} n! \overline{χ^{λ} (1)} = f^{λ}$ , because $χ_{r e g}$ vanishes away from $1$ and $χ^{λ} (1) = \dim_{C} S^{λ} = f^{λ}$ is a nonnegative integer by [F7], so it equals its conjugate. Hence $χ_{r e g} = \sum_{λ ⊢ n} f^{λ} χ^{λ}$ .

3.1F1F8step 2.1step 2.2algebra∎

Applying the linear map $ch$ to step 2.2 and using [F8] gives $p_{1}^{n} = ch (χ_{r e g}) = \sum_{λ ⊢ n} f^{λ} ch (χ^{λ}) = \sum_{λ ⊢ n} f^{λ} s_{λ}$ , the displayed Schur expansion.

The identity is a consistency check of the dictionary: it does not reprove the RSK sum-of-squares formula $\sum_{λ} (f^{λ})^{2} = n!$ .

5 · Examples, counterexamples and false statements

None yet.

Frobenius Characteristic and the Symmetric-Group Character Dictionary

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

The graded ordinary representation ring of the symmetric groups

Definition

The outer induction product of symmetric-group characters

Definition

Complete homogeneous functions expand in power sums with cycle-distribution coefficients

Statement

Facts & Assumptions

Proof

The Frobenius characteristic map

Definition

The Frobenius characteristic is an isometry

Statement

Facts & Assumptions

Proof

The characteristic of a Young permutation character is complete homogeneous

Statement

Facts & Assumptions

Proof

The Frobenius characteristic preserves outer products

Statement

Facts & Assumptions

Proof

The Frobenius characteristic is an isometric graded ring isomorphism

Statement

Facts & Assumptions

Proof

The characteristic of a Specht character is a Schur function

Statement

Facts & Assumptions

Proof

Irreducible symmetric-group character values are power-sum coefficients

Statement

Facts & Assumptions

Proof

Sign twist corresponds to the omega involution

Statement

Facts & Assumptions

Proof

The regular character has characteristic p1np1n​

Statement

Facts & Assumptions

Proof

5 · Examples, counterexamples and false statements

Sources

The regular character has characteristic $p_{1}^{n}$