The dual Jacobi--Trudi identity

The Schur polynomial s_μ is a determinant of elementary symmetric polynomials indexed by the column lengths μ'ⱼ of μ:

s_μ = det (e_{μ'ᵢ - i + j})_{0 ≤ i, j < r}

for every r at least the number of columns of μ, in any number of variables and over any commutative ring. The indices are integers, and e_m is 0 for m < 0; this is TauCeti.esymmInt. Combined with the Jacobi--Trudi identity TauCeti.diagramSchurPoly_eq_det_hsymmInt, it says that the determinant of complete homogeneous symmetric polynomials attached to the rows of μ equals the determinant of elementary symmetric polynomials attached to its columns, YoungDiagram.det_hsymmInt_eq_det_esymmInt; this form holds in any finite alphabet. For a one-column shape the identity is TauCeti.schurPoly_ones, s_{(1ⁿ)} = e_n.

The proof

The route is Macdonald's (I.2.9). Let μ fit in the box with m rows and n columns, and put K = m + n.

• The Toeplitz matrices H = (h_{j - i}) and E = ((-1)^{j - i} e_{j - i}) of size K are inverse to each other (TauCeti.hsymmToeplitz_mul_esymmToeplitz) and have determinant 1.
• The m numbers μ_{m-1-q} + q and the n numbers m + j - μ'ⱼ are complementary in {0, …, K - 1} (Macdonald (I.1.7)): the first family increases in q, the second in j, and m + j - μ'ⱼ < μᵢ + m - 1 - i holds exactly when the cell (i, j) lies in μ, the reverse strict inequality holding otherwise. Listing the first family and then the second is therefore a permutation of {0, …, K - 1}, YoungDiagram.boxShufflePerm, whose inversions are in bijection with the cells of μ, so its sign is (-1)^{|μ|} (YoungDiagram.sign_boxShufflePerm).
• Jacobi's complementary-minor identity for the two Toeplitz matrices, TauCeti.det_submatrix_hsymmToeplitz, applied with the shuffle τ, equates the minor of H on the first m rows and the shuffled first m columns with sign τ times the complementary minor of E. The first is, after reversing the order of its rows and columns, the transpose of the Jacobi--Trudi matrix (h_{μᵢ - i + j}). The second is ((-1)^{μ'ⱼ - j + k} e_{μ'ⱼ - j + k}), whose determinant is (-1)^{|μ|} det (e_{μ'ⱼ - j + k}) by TauCeti.det_submatrix_esymmToeplitz, and the signs cancel.

Main results

• YoungDiagram.det_hsymmInt_eq_det_esymmInt: for μ inside the m × n box, det (h_{μᵢ - i + j})_{i, j < m} = det (e_{μ'ᵢ - i + j})_{i, j < n}.
• TauCeti.diagramSchurPoly_eq_det_esymmInt: the dual Jacobi--Trudi identity for the Schur polynomial of a Young diagram.
• Nat.Partition.schurPoly_eq_det_esymmInt: the dual Jacobi--Trudi identity for the Schur polynomial of a partition.

References

• I. G. Macdonald, Symmetric Functions and Hall Polynomials, Chapter I, (1.7), (2.9), (3.5).
• R. P. Stanley, Enumerative Combinatorics, Vol. 2, Corollary 7.16.2.

The dual Jacobi--Trudi identity

theorem YoungDiagram.det_hsymmInt_eq_det_esymmInt {R : Type u_1} [CommRing R] {σ : Type u_2} [Fintype σ] [DecidableEq σ] (μ : YoungDiagram) {m n : ℕ} (hm : μ.colLen 0 ≤ m) (hn : μ.rowLen 0 ≤ n) : (Matrix.of fun (i j : Fin m) => TauCeti.hsymmInt σ R (↑(μ.rowLen ↑i) - ↑↑i + ↑↑j)).det = (Matrix.of fun (i j : Fin n) => TauCeti.esymmInt σ R (↑(μ.colLen ↑i) - ↑↑i + ↑↑j)).det

The h-determinant of the rows equals the e-determinant of the columns. For a Young diagram μ with at most m rows and at most n columns, det (h_{μᵢ - i + j})_{i, j < m} = det (e_{μ'ᵢ - i + j})_{i, j < n}, where μᵢ and μ'ᵢ are the row and column lengths, and h_k = e_k = 0 for k < 0. By the Jacobi--Trudi identity the left side is the Schur polynomial s_μ, so this is the dual Jacobi--Trudi identity in determinant form.

theorem TauCeti.diagramSchurPoly_eq_det_esymmInt {R : Type u_1} [CommRing R] (N : ℕ) (μ : YoungDiagram) {r : ℕ} (hr : μ.rowLen 0 ≤ r) : diagramSchurPoly N R μ = (Matrix.of fun (i j : Fin r) => esymmInt (Fin N) R (↑(μ.colLen ↑i) - ↑↑i + ↑↑j)).det

The dual Jacobi--Trudi identity. The Schur polynomial of a Young diagram μ in the alphabet Fin N is the determinant det (e_{μ'ᵢ - i + j})_{0 ≤ i, j < r} of elementary symmetric polynomials, where μ'ᵢ is the length of the i-th column, for every r at least the number of columns of μ. The index μ'ᵢ - i + j is an integer, and e_m = 0 for m < 0 (TauCeti.esymmInt).

theorem Nat.Partition.schurPoly_eq_det_esymmInt {R : Type u_1} [CommRing R] {σ : Type u_2} [Fintype σ] {n : ℕ} (μ : n.Partition) {r : ℕ} (hr : (YoungDiagram.ofPartition μ).rowLen 0 ≤ r) : TauCeti.schurPoly σ R μ = (Matrix.of fun (i j : Fin r) => TauCeti.esymmInt σ R (↑((YoungDiagram.ofPartition μ).colLen ↑i) - ↑↑i + ↑↑j)).det

The dual Jacobi--Trudi identity for partitions. In a finite alphabet σ, the Schur polynomial of a partition μ is det (e_{μ'ᵢ - i + j})_{0 ≤ i, j < r}, where μ'ᵢ is the length of the i-th column of the Young diagram of μ, for every r at least the largest part of μ, and e_m = 0 for m < 0.