# garnirElement -- a SpechtModuleElement that is equal to zero

## Synopsis

• Usage:
garnirElement(y,coef,a,b)
garnirElement(y,coef)
garnirElement(y)
• Inputs:
• y, an instance of the type YoungTableau, a tableau that labels a polytabloid
• a, an integer, the row of the descent
• b, , the column of the descent
• coef, an integer, the coefficient of the polytabloid
• Outputs:

## Description

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