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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 $\sum_{λ} (f^{λ})^{2} = n!$ .

Depends on

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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 and §16 (standard reference, not scraped)

The regular character has characteristic p1np1n​