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2013-10-23T20:52:22Z

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In mathematics, the '''Garnir relations''' give a way of expressing a basis of the [[Specht module]]s ''V''λ 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''λ. In characteristic 0, this is an irreducible representation of the [[symmetric group]] ''S''''n''. One can construct ''V''λ explicitly in terms of polytabloids as follows:
* Start with the permutation representation of ''S''''n'' acting on all [[Young tableau]]x of shape λ, where ''S''''n'' 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''''n'' 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''T 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''λ 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''λ. However, the polytabloids associated to different Young tableaux are not necessarily linearly independent, indeed one expects the dimension of ''V''λ to be 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 <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''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:

[[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''''n''. In that case, one can consider the [[Weyl module]]s associated to a partition λ, 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''''n''''. Springer Lecture Notes In Mathematics, 2007.

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

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en>Tomruen at 20:52, 23 October 2013