Specht module: Difference between revisions

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Fix a partition &lambda; of ''n'' and a commutative ring ''k''. ThisThe partition determines a [[Young diagram]] with ''n'' boxes. A [[Young tableau]] of shape &lambda; is a way of labelling the boxes of this Young diagram by distinct numbers <math>1, \dots, n</math>.
A '''tabloid''' is an equivalence class of Young tableaux where two labellings are equivalent if one is obtained from the other by permuting the entries of each row. For each Young tableau ''T'' of shape &lambda; let <math>\{T\}</math> be the corresponding tabloid. The symmetric group on ''n'' points acts on the set of Young tableaux of shape &lambda;. Consequently, it acts on tabloids, and on the vectorfree space''k''-module ''V'' with the tabloids as basis.
Given a Young tableau ''T'' of shape &lambda;, let