In mathematics, a Young symmetrizer is an element of the group algebra of the symmetric group, constructed in such a way that, for the homomorphism from the group algebra to the endomorphisms of a vector space V ⊗ n obtained from the action of S n on V ⊗ n by permutation of indices, the image of the endomorphism determined by that element corresponds to an irreducible representation of the symmetric group over the complex numbers. A similar construction works over any field, and the resulting representations are called Specht modules. The Young symmetrizer is named after British mathematician Alfred Young.
Given a finite symmetric group Sn and specific Young tableau λ corresponding to a numbered partition of n, define two permutation subgroups P λ and Q λ of Sn as follows:
P λ = { g ∈ S n : g preserves each row of λ } and
Q λ = { g ∈ S n : g preserves each column of λ } . Corresponding to these two subgroups, define two vectors in the group algebra C S n as
a λ = ∑ g ∈ P λ e g and
b λ = ∑ g ∈ Q λ sgn ( g ) e g where e g is the unit vector corresponding to g, and sgn ( g ) is the sign of the permutation. The product
c λ := a λ b λ = ∑ g ∈ P λ , h ∈ Q λ sgn ( h ) e g h is the Young symmetrizer corresponding to the Young tableau λ. Each Young symmetrizer corresponds to an irreducible representation of the symmetric group, and every irreducible representation can be obtained from a corresponding Young symmetrizer. (If we replace the complex numbers by more general fields the corresponding representations will not be irreducible in general.)
Let V be any vector space over the complex numbers. Consider then the tensor product vector space V ⊗ n = V ⊗ V ⊗ ⋯ ⊗ V (n times). Let Sn act on this tensor product space by permuting the indices. One then has a natural group algebra representation C S n → End ( V ⊗ n ) on V ⊗ n .
Given a partition λ of n, so that n = λ 1 + λ 2 + ⋯ + λ j , then the image of a λ is
Im ( a λ ) := a λ V ⊗ n ≅ Sym λ 1 V ⊗ Sym λ 2 V ⊗ ⋯ ⊗ Sym λ j V . For instance, if n = 4 , and λ = ( 2 , 2 ) , with the canonical Young tableau { { 1 , 2 } , { 3 , 4 } } . Then the corresponding a λ is given by a λ = e id + e ( 1 , 2 ) + e ( 3 , 4 ) + e ( 1 , 2 ) ( 3 , 4 ) . Let an element in V ⊗ 4 be given by v 1 , 2 , 3 , 4 := v 1 ⊗ v 2 ⊗ v 3 ⊗ v 4 . Then
a λ v 1 , 2 , 3 , 4 = v 1 , 2 , 3 , 4 + v 2 , 1 , 3 , 4 + v 1 , 2 , 4 , 3 + v 2 , 1 , 4 , 3 = ( v 1 ⊗ v 2 + v 2 ⊗ v 1 ) ⊗ ( v 3 ⊗ v 4 + v 4 ⊗ v 3 ) . The latter clearly span Sym 2 V ⊗ Sym 2 V .
The image of b λ is
Im ( b λ ) ≅ ⋀ μ 1 V ⊗ ⋀ μ 2 V ⊗ ⋯ ⊗ ⋀ μ k V where μ is the conjugate partition to λ. Here, Sym i V and ⋀ j V are the symmetric and alternating tensor product spaces.
The image C S n c λ of c λ = a λ ⋅ b λ in C S n is an irreducible representation of Sn, called a Specht module. We write
Im ( c λ ) = V λ for the irreducible representation.
Some scalar multiple of c λ is idempotent, that is c λ 2 = α λ c λ for some rational number α λ ∈ Q . Specifically, one finds α λ = n ! / dim V λ . In particular, this implies that representations of the symmetric group can be defined over the rational numbers; that is, over the rational group algebra Q S n .
Consider, for example, S3 and the partition (2,1). Then one has c ( 2 , 1 ) = e 123 + e 213 − e 321 − e 312
If V is a complex vector space, then the images of c λ on spaces V ⊗ d provides essentially all the finite-dimensional irreducible representations of GL(V).
