Alternants #
For a finite alphabet σ and an exponent vector α : σ → ℕ, the alternant
a_α = det (X_i ^ α_j)_{i, j ∈ σ}
is the determinant of the matrix whose row i records the powers of the variable X_i and whose
column j records a single exponent α_j. It is the antisymmetric counterpart of a monomial
symmetric polynomial: expanding the determinant, a_α is the signed sum of the monomials
∏ᵢ X_{τ i} ^ α_i over the permutations τ of σ.
Alternants are the numerators of Jacobi's bialternant formula s_λ = a_{λ+δ} / a_δ for the Schur
polynomials, and the carriers of Frobenius's formula for the characters of the symmetric group,
p_ν · a_δ = ∑_λ χ^λ(ν) · a_{λ+δ}. The identity that makes the latter compute is the
multiplication rule for power sums proved here,
p_r · a_α = ∑_j a_{α + r e_j},
TauCeti.psum_mul_alternant: multiplying by a power sum raises one exponent at a time. Rewriting
each a_{α + r e_j} in terms of a strictly decreasing exponent vector — it vanishes when an
exponent repeats (TauCeti.alternant_eq_zero_of_not_injective) and otherwise changes by the sign
of the sorting permutation (TauCeti.alternant_comp_perm) — is the move of one bead on the abacus
of beta-numbers, which is how the Murnaghan-Nakayama rule arises from it.
Main definitions #
TauCeti.alternant σ R α: the alternantdet (X_i ^ α_j)inMvPolynomial σ R.
Main statements #
TauCeti.alternant_eq_sum: the Leibniz expansion of the alternant as a signed sum of monomials, andTauCeti.coeff_alternantwithTauCeti.coeff_alternant_of_injectiveits coefficients.TauCeti.alternant_ne_zero_of_injective: an alternant of pairwise distinct exponents is nonzero.TauCeti.isHomogeneous_alternant:a_αis homogeneous of degree∑ i, α i.TauCeti.rename_alternant:a_αis antisymmetric, permuting the variables byemultiplying it bysign e.TauCeti.alternant_comp_permandTauCeti.alternant_eq_zero_of_not_injective: permuting the exponents multipliesa_αby the sign, and a repeated exponent kills it.TauCeti.alternant_add_const: shifting every exponent bycmultipliesa_αby(∏ i, X i) ^ c.TauCeti.alternant_fin_val_eq_vandermonde: for the exponents0, 1, …, n - 1the alternant is the Vandermonde product∏_{i < j} (X_j - X_i).TauCeti.psum_mul_alternant: the power-sum multiplication rulep_r · a_α = ∑_j a_{α + r e_j}.TauCeti.hsymm_mul_alternant: the complete-homogeneous multiplication ruleh_r · a_α = ∑_{|γ| = r} a_{α + γ}.
References #
- I. G. Macdonald, Symmetric Functions and Hall Polynomials, Chapter I, Section 3 (alternants and the bialternant formula) and Section 7 (the characters of the symmetric groups).
- R. P. Stanley, Enumerative Combinatorics, Vol. 2, Sections 7.15 (the classical definition of Schur functions) and 7.17 (the Murnaghan-Nakayama rule).
The alternant a_α = det (X_i ^ α_j) of an exponent vector α : σ → ℕ: the determinant of
the matrix whose entry in row i and column j is the variable X_i raised to the exponent
α_j.
Equations
- TauCeti.alternant σ R α = (Matrix.of fun (i j : σ) => MvPolynomial.X i ^ α j).det
Instances For
The alternant a_α is the determinant of the matrix whose (i, j) entry is X i ^ α j.
The Leibniz expansion of an alternant: a_α is the signed sum, over the permutations τ
of the alphabet, of the monomials ∏ᵢ X_{τ i} ^ α_i.
The coefficient of the monomial x^d in the alternant a_α is the signed count of the
permutations τ that carry d to α, in the sense that d (τ i) = α i for every i.
For an alternant of pairwise distinct exponents, the coefficient of the monomial
∏ᵢ X_i ^ α (τ i) is the sign of τ.
An alternant of pairwise distinct exponents is nonzero: the monomial ∏ᵢ X_i ^ α i
occurs in it with coefficient 1.
An alternant is homogeneous of degree the total of its exponents.
Alternants are antisymmetric: renaming the variables along a permutation e multiplies
an alternant by the sign of e.
Permuting the exponents of an alternant multiplies it by the sign of the permutation.
An alternant with a repeated exponent vanishes.
Raising every exponent of an alternant by c multiplies it by (∏ i, X i) ^ c.
The Vandermonde alternant: for the exponents 0, 1, …, n - 1 the alternant is the
Vandermonde product ∏_{i < j} (X_j - X_i).
The power-sum multiplication rule for alternants: multiplying a_α by the power sum
p_r = ∑ᵢ X_i ^ r gives the sum of the alternants obtained from α by raising a single exponent
by r, p_r · a_α = ∑_j a_{α + r e_j}.
The complete-homogeneous multiplication rule for alternants: multiplying a_α by the
complete homogeneous symmetric polynomial h_r gives the sum of the alternants obtained from α by
adding an exponent vector of total degree r, h_r · a_α = ∑_{|γ| = r} a_{α + γ}.
Unlike the power-sum rule, the shifts here are not supported at a single index, so the terms with a repeated exponent do not account for all the cancellation: the surviving terms still have to be sorted back into decreasing order. That extra step is what separates the Pieri rule from the Murnaghan-Nakayama rule.