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.
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Definition
Fix a partition λ of n. A tabloid is an equivalence class of labellings of the Young diagram of shape λ, where two labellings are equivalent if one is obtained from the other by permuting the entries of each row. Denote by
where QT is the subgroup of permutations, preserving (as sets) all columns of T and
The Specht module has a basis of elements ET for T a standard Young tableau.
A gentle introduction to the construction of the Specht module may be found in Section 1 of "Specht Polytopes and Specht Matroids".
Structure
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.