In mathematics, a Specht module is one of the representations of symmetric groups 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 of n and a commutative ring k. The partition determines a Young diagram with n boxes. A Young tableau of shape .

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 be the corresponding tabloid. The symmetric group on n points acts on the set of Young tableaux of shape . Consequently, it acts on tabloids, and on the free k-module V with the tabloids as basis.

Given a Young tableau T of shape , let

where QT is the subgroup of permutations preserving (as sets) all columns of T is the sign of the permutation . 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.

A gentle introduction to the construction of the Specht module may be found in Section 1 of "Specht Polytopes and Specht Matroids".

Structure

is the number of standard Young tableaux. It is given by the hook length formula.

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 (for a prime number p) 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.

See also

• Garnir relations, a more detailed description of the structure of Specht modules.

References