generalizedVandermondeMatrix(List,List,PolynomialRing) -- the method for calculating generalized Vandermonde matrices

Synopsis

Function: generalizedVandermondeMatrix
Usage:

generalizedVandermondeMatrix(indices,exponents,R)
Inputs:
- indices, a list, a list of the variables that appear in each column of the matrix
- exponents, a list, a list of the powers that appear in each row of the matrix
- R, a polynomial ring,
Outputs:
- a matrix,

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