### 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. rect.matrixcell { fill: #FFFFC0; stroke: #A0A0A0; stroke-width: 1; } rect.matrixdeletedcell { fill: #A0A0A0; stroke: #A0A0A0; stroke-width: 1; } text.matrixentry { font-family: Serif; font-weight: normal; font-style: normal; dominant-baseline: auto; text-anchor: middle; } 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:  =  rect.lambdamucell { fill: #30A030; stroke: none; } By the Jacobi–Trudi formula, sλ/μ = det [hλ(j)−μ(k)+k−j]1 ≤ j,k ≤ len(λ), i.e. sλ/μ can be computed as the determinant of the following matrix: rect.completecell { fill: #FFFFC0; stroke: #A0A0A0; stroke-width: 1; } text.completeentry { font-family: Serif; font-weight: normal; font-style: normal; dominant-baseline: auto; text-anchor: middle; } 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)+k−j]1 ≤ j,k ≤ len(λ'), i.e. sλ/μ equals the determinant of the following matrix constructed from λ' and μ'. rect.theoreticalcell { fill: #FFFFC0; stroke: #A0A0A0; stroke-width: 1; } text.theoreticalentry { font-family: Serif; font-weight: normal; font-style: normal; dominant-baseline: auto; text-anchor: middle; }

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