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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 ''V''λ explicitly in terms of polytabloids as follows: * Start with the permutation representation of ''S''''n'' acting on all Young tableaux 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 tabloids, which are orbits of Young tableaux under the action of the Young row subgroups (two Young tableaux of shape λ, where , are equivalent if they are in the same orbit of , acting by permuting the entries in each row). * Now consider polytabloids, 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 , where 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: : * 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Garnir relations」の詳細全文を読む スポンサード リンク
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