The finite-variable Frobenius characteristic and Frobenius's formula #
For a finite alphabet σ of at least n letters, the complex class functions of Sₙ and the
symmetric homogeneous polynomials of degree n in the variables σ have bases indexed by the
partitions of n. The first basis consists of the characters of the complex Specht modules; the
second consists of the Schur polynomials. The Frobenius characteristic is the linear equivalence
taking one basis to the other.
Its classical description is the cycle-type formula: the Frobenius characteristic of a class
function f is the average of the power-sum products over the cycle types,
ch(f) = (1 / n!) · ∑_{π ∈ Sₙ} f(π) · p_{ρ(π)}.
That formula rests on Frobenius's formula, which expands the power-sum products in the Schur
polynomials with the character table of Sₙ as the matrix of coefficients,
p_ρ = ∑_{λ ⊢ n} χ^λ(ρ) · s_λ,
and on its inverse ∑_{π ∈ Sₙ} χ^μ(π) · p_{ρ(π)} = n! · s_μ. Both are proved here, in any finite
alphabet and over any commutative ring, through the monomial symmetric polynomials m_ν. The
power sums expand as p_ρ = ∑_ν ψ^ν(ρ) m_ν with the permutation characters ψ^ν as coefficients
(TauCeti.psumPart_eq_sum_card_fixedPoints_smul_msymm), and the Schur polynomials as
s_λ = ∑_ν K_{λν} m_ν with the Kostka numbers (TauCeti.schurPoly_eq_sum_kostkaNumber_smul_msymm).
Young's rule ψ^ν = ∑_λ K_{λν} χ^λ
(TauCeti.char_permutationModule_eq_sum_kostkaNumber_mul_spechtChar) converts the first expansion
into the second, which is Frobenius's formula; its character-pairing form
∑_π χ^μ(π) ψ^ν(π) = n! K_{μν} (TauCeti.sum_spechtChar_mul_char_permutationModule) gives the
inverse directly, without passing through the orthogonality relations.
Main definitions #
TauCeti.frobeniusCharacteristic: the basis-preserving linear equivalence to symmetric homogeneous polynomials.
Main results #
TauCeti.frobeniusCharacteristic_spechtCharacter: the image of the character ofS^μis the Schur polynomials_μ.TauCeti.schurPolyBasis_repr_frobeniusCharacteristic: each Schur coordinate is the pairing with the corresponding Specht character.TauCeti.psumPart_eq_sum_spechtCharValue_smul_schurPoly: Frobenius's formulap_ρ = ∑_λ χ^λ(ρ) s_λ.TauCeti.sum_spechtChar_smul_psumPart: its inverse,∑_π χ^μ(π) p_{ρ(π)} = n! s_μ.TauCeti.frobeniusCharacteristic_eq_sum_psumPart: the cycle-type formula for the Frobenius characteristic.TauCeti.hallInner_psumPart_schurPolyBasis: for the Hall inner productTauCeti.hallInner,⟨p_ρ, s_μ⟩ = χ^μ(ρ).TauCeti.hallInner_psumPart: the power sums are orthogonal,⟨p_ρ, p_ρ'⟩ = z_ρ δ_{ρρ'}.TauCeti.hallInner_frobeniusCharacteristic: the Frobenius characteristic is an isometry from the character pairing to the Hall inner product.
References #
- I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Chapter I, Section 7,
where the power sums are expanded in the Schur functions with the characters of
Sₙas coefficients and the characteristic map is shown to be an isometry; and Section 4 for the orthogonality of the power sums. - W. Fulton and J. Harris, Representation Theory: A First Course (1991), Lecture 4, §4.1 (Frobenius's formula) and Appendix A.
The finite-variable Frobenius characteristic, for an alphabet σ of at least n letters, is
the linear equivalence sending the character of each complex Specht module to the Schur polynomial
of the same shape.
Instances For
The Frobenius characteristic takes an irreducible character to its Schur polynomial.
The Schur coordinate of a Frobenius characteristic is the pairing with the corresponding Specht character.
The Schur coefficient of a class function under the Frobenius characteristic is its character pairing with the corresponding Specht character. This gives an explicit formula for the map on arbitrary class functions.
Frobenius's formula #
Frobenius's formula: the power-sum product of a partition ρ of n expands in the Schur
polynomials as p_ρ = ∑_{λ ⊢ n} χ^λ(ρ) s_λ, the coefficients being the column of ρ in the
character table of Sₙ. This holds in every finite alphabet and over every commutative ring; the
Schur polynomials of the partitions with more parts than the alphabet has letters vanish.
The inverse of Frobenius's formula: weighting the power-sum products over the cycle types by
a Specht character recovers the Schur polynomial, ∑_{π ∈ Sₙ} χ^μ(π) p_{ρ(π)} = n! s_μ. Over a
ℚ-algebra this says that (1 / n!) ∑_π χ^μ(π) p_{ρ(π)} is s_μ.
The cycle-type formula #
The cycle-type formula for the Frobenius characteristic: the Frobenius characteristic of a
class function f of Sₙ is (1 / n!) ∑_{π ∈ Sₙ} f(π) p_{ρ(π)}, the average of the power-sum
products over the cycle types weighted by f.
The Hall inner product #
A power sum pairs with a Schur polynomial to a character value: ⟨p_ρ, s_μ⟩ = χ^μ(ρ) for
the Hall inner product, over every commutative ring. This is Frobenius's formula
p_ρ = ∑_λ χ^λ(ρ) s_λ read in the Schur basis. It is not a simp lemma:
TauCeti.hallInner_schurPolyBasis_right already reduces the left side to a Schur coordinate.
The power sums are orthogonal for the Hall inner product: in an alphabet of at least n
letters, ⟨p_ρ, p_ρ'⟩ = z_ρ δ_{ρρ'}, where z_ρ = ∏ᵢ i^{mᵢ} mᵢ! is the order of the centralizer
of a permutation of cycle type ρ (TauCeti.zPart). This holds over every commutative ring: by
TauCeti.hallInner_psumPart_schurPolyBasis it is the column orthogonality of the character table
of Sₙ. With fewer letters the formula can fail: the family of the power sums of all partitions
of n is then larger than the Schur basis, so it is linearly dependent over a nontrivial ring.
The Frobenius characteristic is an isometry: it carries the character pairing
⟨f, g⟩ = (1 / n!) ∑_π f(π) g(π⁻¹) of class functions of Sₙ to the Hall inner product of
symmetric polynomials. Both forms make the bases matched by the Frobenius characteristic
orthonormal: the characters of the Specht modules and the Schur polynomials.