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

## Synopsis

• Function: spechtPolynomial
• Usage:
spechtPolynomial(y,R)
• Inputs:
• Optional inputs:
• AsExpression => ..., default value false, an optional argument that returns polynomials as expressions
• Outputs:
• , 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