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 $i$th column and a subset $B$ of the $i+1$ column. It is defined as $ \sum_{\pi} sgn(\pi)\pi(T)$. The $\pi$ are called transversals. They are a set of permutations such that $S_{A \cup B}$ is the disjoint union of $\pi(S_A \times S_B)$.
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 $e_T$ can be written as $ e_T = -\sum_{\pi \neq id} sgn(\pi)\pi(e_T) $
In this implementation the $i$th column is taken to be the parameter b. The set $A$ is all the cells in the $i$th 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 |
The object garnirElement is a method function.