Specht module

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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.

Definition[edit]

Fix a partition λ of n. A tabloid is an equivalence class of labelings of the Young diagram of shape λ, where two labelings are equivalent if one is obtained from the other by permuting the entries of each row. The symmetric group on n points acts on the set of tabloids, and therefore on the module V with the tabloids as basis. For each Young tableau T of shape λ, form the element

E_T=\sum_{\sigma\in Q_T}\epsilon(\sigma)\sigma(T)

where QT is the subgroup fixing all columns of T, and ε is the sign of a permutation. 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 λ is the module generated by the elements ET as T runs through all tableaux of shape λ.

The Specht module has a basis of elements ET for T a standard Young tableau.

Structure[edit]

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.

References[edit]