Complementary minors of the h and e Toeplitz matrices

The Toeplitz matrix (h_{j - i}) of complete homogeneous symmetric polynomials and the signed Toeplitz matrix ((-1)^{j - i} e_{j - i}) of elementary ones are inverse to each other (TauCeti.hsymmToeplitz_mul_esymmToeplitz), and both are upper unitriangular, so both have determinant 1. Jacobi's complementary-minor identity (Matrix.det_submatrix_comp_inr_eq_sign_mul_det_submatrix_comp_inl_of_det_eq_one) therefore reads every minor of the first off the complementary minor of the second, with no factor beyond the sign of the two permutations that cut the indices in two.

This is the determinantal half of the passage from the Jacobi--Trudi identity TauCeti.diagramSchurPoly_eq_det_hsymmInt, a determinant of complete homogeneous symmetric polynomials of size the number of rows of a shape, to its dual (transpose) form, a determinant of elementary ones of size the number of columns: the two matrices are complementary minors of the two Toeplitz matrices, for the permutations that read off the beta-numbers of a partition and of its conjugate.

Combined with TauCeti.det_submatrix_esymmToeplitz, which pulls the signs out of a minor of the signed Toeplitz matrix, this expresses the minor of the h Toeplitz matrix through a minor of the unsigned integer-indexed family TauCeti.esymmInt, the shape a dual Jacobi--Trudi determinant has.

Main results

• TauCeti.det_submatrix_hsymmToeplitz: a minor of the h Toeplitz matrix is the complementary minor of the signed e Toeplitz matrix, up to the sign of the two permutations.

References

• I. G. Macdonald, Symmetric Functions and Hall Polynomials, Chapter I, Section 2, Example 3, and Section 3, formula (3.5).

theorem TauCeti.det_submatrix_hsymmToeplitz (σ : Type u_1) [Fintype σ] [DecidableEq σ] (R : Type u_2) [CommRing R] {m : Type u_3} {p : Type u_4} [Fintype m] [Fintype p] [DecidableEq m] [DecidableEq p] (N : ℕ) (e : m ⊕ p ≃ Fin N) (π ρ : Equiv.Perm (Fin N)) : ((hsymmToeplitz σ R N).submatrix (⇑ρ ∘ ⇑e ∘ Sum.inr) (⇑π ∘ ⇑e ∘ Sum.inr)).det = ↑↑(Equiv.Perm.sign π) * ↑↑(Equiv.Perm.sign ρ) * ((esymmToeplitz σ R N).submatrix (⇑π ∘ ⇑e ∘ Sum.inl) (⇑ρ ∘ ⇑e ∘ Sum.inl)).det

Jacobi's complementary-minor identity for the two Toeplitz matrices of symmetric polynomials. A minor of (h_{j - i}) on the rows ρ (p) and the columns π (p) is the complementary minor of ((-1)^{j - i} e_{j - i}), on the rows π (m) and the columns ρ (m), signed by π and ρ.