Specht module
Encyclopedia
In mathematics, a Specht module is one of the representations of symmetric group
s studied by .
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.
A tabloid is an equivalence class of Young tableaux of shape λ, where two tableaux 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 tableau T, form the element
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.
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.
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
s studied by .
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
Fix a partition λ of n.A tabloid is an equivalence class of Young tableaux of shape λ, where two tableaux 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 tableau T, form the element
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
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.