schurPolynomial(List,Partition,PolynomialRing) -- a method for constructing Schur polynomials

Synopsis

• Function: schurPolynomial
• Usage:
schurPolynomial(indices,parti,R)
• Inputs:
• indices, a list, a list of the variables that appear in each column of the matrix
• parti, an instance of the type Partition, a partition that indexes the schur polynomial
• R, ,
• Optional inputs:
• AsExpression => ..., default value false, an optional argument that returns polynomials as expressions
• Strategy (missing documentation) => ..., default value semistandard_tableaux,
• Outputs:
• ,

Description

Generalized vandermonde matrices allow the power in the rows to be different from the numbers from 0 to n-1.

 i1 : R = QQ[x_0..x_4] o1 = R o1 : PolynomialRing i2 : M = generalizedVandermondeMatrix({0,2,3},{1,3,5},R) o2 = | x_0 x_2 x_3 | | x_0^3 x_2^3 x_3^3 | | x_0^5 x_2^5 x_3^5 | 3 3 o2 : Matrix R <--- R

The determinant of these matrices divided by the Vandermonde determinant of the same rank is equal to a schur polynomial .

 i3 : (determinant M)//vandermondeDeterminant({0,2,3},R) 3 2 2 3 3 2 2 2 2 3 2 2 3 2 3 o3 = x x x + x x x + x x x + 2x x x + x x x + x x x + x x x 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 0 2 3 o3 : R