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 symmetric group on n points acts on the set of Young tableaux of shape λ. Consequently, it acts on tabloids, and on the free k-module V with the tabloids as basis.
Given a Young tableau T of shape λ, let where QT is the subgroup of permutations, preserving (as sets) all columns of T and
The Specht module has a basis of elements ET for T a standard Young tableau.
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.