Garnir relations

In mathematics, the Garnir relations give a way of expressing a basis of the Specht modules "V"λ in terms of standard polytabloids.
Specht modules in terms of polytabloids.
Given a partition "λ" of "n", one has the Specht module "V"λ. In characteristic 0, this is an irreducible representation of the symmetric group "S""n". One can construct "Vλ" explicitly in terms of polytabloids as follows:
Straightening polytabloids and the Garnir elements.
The above construction gives an explicit description of the Specht module "V"λ. However, the polytabloids associated to different Young tableaux are not necessarily linearly independent, indeed, the dimension of "V"λ is exactly the number of standard Young tableaux of shape λ. In fact, the polytabloids associated to standard Young tableaux span "V"λ; to express other polytabloids in terms of them, one uses a straightening algorithm.
Given a Young tableau "S", we construct the polytabloid "e""S" as above. Without loss of generality, all columns of "S" are increasing, otherwise we could instead start with the modified Young tableau with increasing columns, whose polytabloid will differ at most by a sign. "S" is then said to not have any "column descents". We want to express "e""S" as a linear combination of standard polytabloids, i.e. polytabloids associated to standard Young tableaux. To do this, we would like permutations π"i" such that in all tableaux "S"πi, a row descent has been eliminated, with formula_5. This then expresses "S" in terms of polytabloids that are closer to being standard. The permutations that achieve this are the Garnir elements.
Suppose we want to eliminate a row descent in the Young tableau "T". We pick two subsets "A" and "B" of the boxes of "T" as in the following diagram:
Then the Garnir element formula_6 is defined to be formula_7, where the π"i" are the permutations of the entries of the boxes of "A" and "B" that keep both subsets "A" and "B" without column descents.
Example.
Consider the following Young tableau:
There is a row descent in the second row, so we choose the subsets "A" and "B" as indicated, which gives us the following:
This gives us the Garnir element formula_8. This allows us to remove the row descent in the second row, but this has also introduced other descents in other places. But there is a way in which all tableaux obtained like this are closer to being standard, this is measured by a "dominance order" on polytabloids. Therefore, one can repeatedly apply this procedure to "straighten" a polytabloid, eventually writing it as a linear combination of standard polytabloids, showing that the Specht module is spanned by the standard polytabloids. As they are also linearly independent, they form a basis of this module.
Other interpretations.
There is a similar description for the irreducible representations of "GL""n". In that case, one can consider the Weyl modules associated to a partition λ, which can be described in terms of bideterminants. One has a similar straightening algorithm, but this time in terms of semistandard Young tableaux.