spechtPolynomial(YoungTableau,PolynomialRing) -- the Specht polynomial indexed by a standard tableau

Synopsis

Function: spechtPolynomial
Usage:

spechtPolynomial(y,R)
Inputs:
- y, an instance of the type YoungTableau,
- R, a polynomial ring,
Optional inputs:
- AsExpression => ..., default value false, an optional argument that returns polynomials as expressions
Outputs:
- a ring element, the Specht polynomial

Description

Specht polynomials were the original objects that gave rise to the Specht modules. The Specht polynomial of a tableau $T$ is product of the Vandermonde determinant of the variables index by the columns of the tableau.

i1 : R = QQ[x_0..x_4]

o1 = R

o1 : PolynomialRing

i2 : p = new Partition from {2,2,1}

o2 = Partition{2, 2, 1}

o2 : Partition

i3 : y = youngTableau(p,{0,3,1,4,2})

o3 = | 0 3 |
     | 1 4 |
     | 2 |

o3 : YoungTableau

i4 : spechtPolynomial(y,R)

      2          2      2        2          2        2      2          2    
o4 = x x x  - x x x  - x x x  + x x x  + x x x  - x x x  - x x x  + x x x  +
      0 1 3    0 1 3    0 2 3    1 2 3    0 2 3    1 2 3    0 1 4    0 1 4  
     ------------------------------------------------------------------------
      2        2          2        2
     x x x  - x x x  - x x x  + x x x
      0 2 4    1 2 4    0 2 4    1 2 4

o4 : R

i5 : factor oo

o5 = (x  - x )(x  - x )(x  - x )(x  - x )
       3    4   1    2   0    2   0    1

o5 : Expression of class Product