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.
Corresponding to these two subgroups, define two vectors in the group algebra as
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. (If we replace the complex numbers by more general fields the corresponding representations will not be irreducible in general.)
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 (2,1). 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(V).