# 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 *S*_{n} and specific Young tableau λ corresponding to a numbered partition of *n*, define two permutation subgroups and of *S*_{n} 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 signature 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.)

## Construction

Let *V* be any vector space over the complex numbers. Consider then the tensor product vector space (*n* times). Let *S*_{n} 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

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^{[1]} of *S*_{n}, 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, *S*_{3} 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).

## See also

## Notes

- ↑ See Template:Harv

## References

- William Fulton.
*Young Tableaux, with Applications to Representation Theory and Geometry*. Cambridge University Press, 1997. - Lecture 4 of Template:Fulton-Harris
- Bruce E. Sagan.
*The Symmetric Group*. Springer, 2001.