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In mathematics, the '''Garnir relations''' give a way of expressing a basis of the [[Specht module]]s ''V''<sub>λ</sub> in terms of standard polytabloids.

==Specht modules in terms of polytabloids==
Given a [[partition (number theory)|partition]] λ of ''n'', one has the Specht module ''V''<sub>λ</sub>. In characteristic 0, this is an irreducible representation of the [[symmetric group]] ''S''<sub>''n''</sub>. One can construct ''V''<sub>λ</sub> explicitly in terms of polytabloids as follows:
* Start with the permutation representation of ''S''<sub>''n''</sub> acting on all [[Young tableau]]x of shape λ, where ''S''<sub>''n''</sub> acts by permuting the entries in each tableau. Note that we do not require the tableaux to be standard.
* Extend this to an action of ''S''<sub>''n''</sub> on all (row) [[Young tabloid]]s, which are orbits of Young tableaux under the action of the Young row subgroups (two Young tableaux of shape λ, where <math>\lambda = \lambda_1 + \cdots + \lambda_r</math>, are equivalent if they are in the same orbit of <math>S_{\lambda_1} \times \cdots \times S_{\lambda_r} \leqslant S_n</math>, acting by permuting the entries in each row).
* Now consider [[polytabloid]]s, these are formal linear combinations of Young tabloids, with integer coefficients. Given any Young tableau ''T'', one defines the associated polytabloid by acting on ''T'' with the Young column subgroup <math>S_{\mu_1} \times \cdots \times S_{\mu_s}</math>, where <math>\mu = \mu_1 + \cdots + \mu_s</math> is the [[conjugate partition]] to λ. One writes a polytabloid ''S'' = ''T'' σ corresponding to each element in this orbit, affected with the sign of the permutation σ taking ''T'' to ''S''. One then writes ''e''<sub>T</sub> for the corresponding polytabloid:
:<math>e_T = \sum_{\sigma \in S_{\mu_1} \times \cdots \times S_{\mu_s}} \sgn(\sigma) T \sigma. </math>
* The Specht module ''V''<sub>λ</sub> is then the subspace of the space of all polytabloids spanned by the polytabloids obtained from Young tableaux in the above fashion.

==Straightening polytabloids and the Garnir elements==
The above construction gives an explicit description of the Specht module ''V''<sub>λ</sub>. However, the polytabloids associated to different Young tableaux are not necessarily linearly independent, indeed one expects the dimension of ''V''<sub>λ</sub> to be exactly the number of '''standard''' Young tableaux of shape λ. In fact, the polytabloids associated to standard Young tableaux span ''V''<sub>λ</sub>; to express other polytabloids in terms of them, one uses a '''straightening algorithm'''.

Given a Young tableau ''S'', we construct the polytabloid ''e''<sub>''S''</sub> 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''<sub>''S''</sub> as a linear combination of standard polytabloids, i.e. polytabloids associated to standard Young tableaux. To do this, we would like permutations π<sub>''i''</sub> such that in all tableaux ''S''π<sub>i</sub>, a row descent has been eliminated, with <math> S(1 + \sum_i \pi_i) = 0</math>. 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:

[[File:Row descent.svg]]

Then the Garnir element <math>g_{A,B}</math> is defined to be <math>\sum_i \sgn(\pi_i) \pi_i</math>, where the π<sub>''i''</sub> 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:

[[File:Row descent and Garnir element.svg]]

There is a row descent in the second row, so we choose the subsets ''A'' and ''B'' as indicated, which gives us the following:

[[File:Straightening of a polytabloid.svg]]

This gives us the Garnir element <math>g_{A,B} = 1 - (4 5) + (2 4 5) + (4 6 5) - (2 4 6 5) + (2 5) (4 6)</math>. 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''<sub>''n''</sub>. In that case, one can consider the [[Weyl module]]s associated to a partition &lambda;, which can be described in terms of [[bideterminant]]s. One has a similar straightening algorithm, but this time in terms of semistandard Young tableaux.

==References==
* William Fulton. ''Young Tableaux, with Applications to Representation Theory and Geometry''. Cambridge University Press, 1997. 
* [[Bruce Sagan|Bruce E. Sagan]]. ''The Symmetric Group''. Springer, 2001.
* Sandy Green. ''Polynomial Representations of ''GL''<sub>''n''</sub>''. Springer Lecture Notes In Mathematics, 2007.

[[Category:Algebraic combinatorics]]
[[Category:Representation theory]]
[[Category:Representation theory of finite groups]]