Minors of band Toeplitz matrices and skew Schur polynomials: interactive illustration

Egor Maximenko and Mario Alberto Moctezuma Salazar, 2017-05-24.
This program illustrates Theorem 2.1 from our paper. For simplicity, we omit the signs.
This result follows easily from Per Alexandersson's paper, formula (3).
See also tests in SageMath.


     n = 10 (the order of the matrix)

     p = 3 (the number of diagonals below the main diagonal)

Click near the matrix to delete or retrieve rows and columns.

Deleted rows: .
Deleted columns: .

Submatrix order:  m = .
 

The Toeplitz minor shown on the left can be expressed as the skew Schur polynomial sλ/μ,
where λ and μ are the following integer partitions:

λ = ,

μ = .

They form the following skew partition and are represented by the following diagram:

 = 

By the Jacobi–Trudi formula, sλ/μ = det [hλ(j)−μ(k)+kj]1 ≤ j,k ≤ len(λ),
i.e. sλ/μ can be computed as the determinant of the following matrix:

Let us verify that the formula is correct in this example.
Compute the conjugate skew partition corresponding to the transposed diagram:

λ' = ,   μ' = .

By the dual Jacobi–Trudi formula, sλ/μ = det [eλ'(j)−μ'(k)+kj]1 ≤ j,k ≤ len(λ'),
i.e. sλ/μ equals the determinant of the following matrix constructed from λ' and μ'.




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