In mathematics, the Garnir relations give a way of expressing a basis of the Specht modules Vλ in terms of standard polytabloids.
Given an integer partition λ of n, one has the Specht module Vλ.
In characteristic 0, this is an irreducible representation of the symmetric group Sn.
However, the polytabloids associated to different Young tableaux are not necessarily linearly independent, indeed, the dimension of Vλ is exactly the number of standard Young tableaux of shape λ.
To do this, we would like permutations πi such that in all tableaux Sπi, a row descent has been eliminated, with
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:
, where the πi are the permutations of the entries of the boxes of A and B that keep both subsets A and B without column descents.
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.
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.