# spechtPolynomials(Partition,PolynomialRing) -- the set of all Specht polynomial indexed by standard tableaux of shape p

## Synopsis

• Function: spechtPolynomials
• Usage:
spechtPolynomials(p,R)
• Inputs:
• Optional inputs:
• AsExpression => ..., default value false, an optional argument that returns polynomials as expressions
• Outputs:
• , a hash table with the polynomials index by the filling of their respective tableaux

## Description

The set of all the Specht polynomials for standard tableaux of a given shape p forms a basis for a module which is isomorphich to the Specht module indexed by p.

 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 : specht = spechtPolynomials(p,R) 2 2 2 2 2 2 2 2 2 2 2 2 o3 = HashTable{{0, 1, 2, 3, 4} => x x x - x x x - x x x + x x x - 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 2 0 1 2 0 2 3 0 2 3 0 1 4 1 2 4 0 3 4 2 3 4 0 1 4 1 2 4 0 3 4 2 3 4 2 2 2 2 2 2 2 2 2 2 2 2 {0, 1, 2, 4, 3} => x x x - x x x - x x x + x x x + 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 2 0 1 2 0 1 3 1 2 3 0 1 3 1 2 3 0 2 4 0 2 4 0 3 4 2 3 4 0 3 4 2 3 4 2 2 2 2 2 2 2 2 2 2 2 2 {0, 2, 1, 3, 4} => x x x - x x x - x x x + x x x - 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 2 0 1 2 0 1 3 0 1 3 0 2 4 1 2 4 0 3 4 1 3 4 0 2 4 1 2 4 0 3 4 1 3 4 2 2 2 2 2 2 2 2 2 2 2 2 {0, 2, 1, 4, 3} => x x x - x x x - x x x + x x x + 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 2 0 1 2 0 2 3 1 2 3 0 2 3 1 2 3 0 1 4 0 1 4 0 3 4 1 3 4 0 3 4 1 3 4 2 2 2 2 2 2 2 2 2 2 2 2 {0, 3, 1, 4, 2} => x x x - x x x - x x x + x x x + 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 0 2 4 1 2 4 0 2 4 1 2 4 o3 : HashTable