# 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 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".[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.