High Quality Content by WIKIPEDIA articles! In mathematics, a Specht module is one of the representations of symmetric groups studied by Wilhelm Specht (1935). They are indexed by partitions, and in characteristic 0 the Specht modules of partitions of n form a complete set of irreducible representations of the symmetric group on n points. The elements ET can be considered as elements of the module V, by mapping each tableau to the tabloid it generates. The Specht module of the partition ? is the module generated by the elements ET as T runs through all tableaux of shape ?. The Specht module has a basis of elements ET for T a standard Young tableau. Over fields of characteristic 0 the Specht modules are irreducible, and form a complete set of irreducible representations of the symmetric group. A partition is called p-regular if it does not have p parts of the same (positive) size. Over fields of characteristic p>0 the Specht modules can be reducible. For p-regular partitions they have a unique irreducible quotient, and these irreducible quotients form a complete set of irreducible representations.

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