Trisha Shetty (Editor)

Young symmetrizer

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

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.

Contents

Definition

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.)

Construction

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).

References

Young symmetrizer Wikipedia


Similar Topics