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

Synopsis

Function: spechtPolynomials
Usage:

spechtPolynomials(p,R)
Inputs:
- p, an instance of the type Partition,
- R, a polynomial ring,
Optional inputs:
- AsExpression => ..., default value false, an optional argument that returns polynomials as expressions
Outputs:
- a hash table, 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 isomorphic 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

Ways to use this method: