Young symmetrizer


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 obtained from the action of on 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.

Definition

Given a finite symmetric group Sn and specific Young tableau λ corresponding to a numbered partition of n, define two permutation subgroups and of Sn as follows:
and
Corresponding to these two subgroups, define two vectors in the group algebra as
and
where is the unit vector corresponding to g, and is the sign of the permutation. The product
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.

Construction

Let V be any vector space over the complex numbers. Consider then the tensor product vector space . Let Sn act on this tensor product space by permuting the indices. One then has a natural group algebra representation on.
Given a partition λ of n, so that, then the image of is
For instance, if, and, with the canonical Young tableau. Then the corresponding is given by
Let an element in be given by. Then
The latter clearly span
The image of is
where μ is the conjugate partition to λ. Here, and are the symmetric and alternating tensor product spaces.
The image of in is an irreducible representation of Sn, called a Specht module. We write
for the irreducible representation.
Some scalar multiple of is idempotent, that is for some rational number Specifically, one finds. In particular, this implies that representations of the symmetric group can be defined over the rational numbers; that is, over the rational group algebra.
Consider, for example, S3 and the partition. Then one has
If V is a complex vector space, then the images of on spaces provides essentially all the finite-dimensional irreducible representations of GL.