A Garnir element is an element which is constructed to remove row descents from a tableau. Given a tableau T, the Garnir element is defined for a subset A of the ith column and a subset B of the i+1 column. It is defined as ∑π sgn(π)π(T). The π are called transversals. They are a set of permutations such that SA ∪B is the disjoint union of π(SA ×SB).
The identity can always be chosen as a transversal for any pair of sets. Therefore the original tableau T appears along side other tableaux which are closer to being standard. Another property is that this element is equal to zero. Therefore the original polytabloid eT can be written as eT = -∑π≠id sgn(π)π(eT)
In this implementation the ith column is taken to be the parameter b. The set A is all the cells in the ith column from the a-th row to the bottom. The set B is all the cells in the i+1 column from the a-th row to the top.
If the number (a,b) are not specified then they are taken as the coordinates of the first row descent of T
i1 : p = new Partition from {3,2,1} o1 = Partition{3, 2, 1} o1 : Partition |
i2 : y = youngTableau(p,{1,2,3,5,4,6}) o2 = | 1 2 3 | | 5 4 | | 6 | o2 : YoungTableau |
i3 : garnirElement y o3 = - | 1 2 3 | + | 1 2 3 | + | 1 4 3 | - | 1 4 3 | + | 1 5 3 | + | 1 2 3 | | 4 5 | | 4 6 | | 2 5 | | 2 6 | | 2 6 | | 5 4 | | 6 | | 5 | | 6 | | 5 | | 4 | | 6 | o3 : SpechtModuleElement |