The index monomial is used in the construction of higher Specht polynomials. To calculate the index monomial first the index tableau of $S$, $i(S)$ is calculated. Then the monomial is calculated as $x_T^{i(S)}$. This is a monomial with the variables as they appear in T with the exponents that appear in $i(S)$.
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 : S = youngTableau(p,{0,2,1,3,4})
o3 = | 0 2 |
| 1 3 |
| 4 |
o3 : YoungTableau
|
i4 : T = youngTableau(p,{0,1,2,3,4})
o4 = | 0 1 |
| 2 3 |
| 4 |
o4 : YoungTableau
|
i5 : ind = indexTableau(S)
o5 = | 0 1 |
| 1 2 |
| 3 |
o5 : YoungTableau
|
i6 : indexMonomial(S,T,R)
2 3
o6 = x x x x
1 2 3 4
o6 : R
|