# Specht module

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 Specht module has a basis of elements *E*_{T} 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".^{[1]}

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.