Young symmetrizer

Short description: Element of the group algebra of a symmetric group

In mathematics, a Young symmetrizer is an element of the group algebra of the symmetric group Sn whose natural action on tensor products V⊗n of a complex vector space V has as image an irreducible representation of the group of invertible linear transformations GL(V). All irreducible representations of GL(V) are thus obtained. It is constructed from the action of Sn on the vector space V⊗n by permutation of the different factors (or equivalently, from the permutation of the indices of the tensor components). A similar construction works over any field, but in positive characteristic (in particular, over finite fields) the image need not be an irreducible representation. The Young symmetrizers also act on the vector space of functions on Young tableau and the resulting representations are called Specht modules which again construct all complex irreducible representations of the symmetric group while the analogous construction in prime characteristic need not be irreducible. The Young symmetrizer is named after British mathematician Alfred Young.

## Contents

* 1 Definition
* 2 Construction
* 3 See also
* 4 Notes
* 5 References

## Definition

Given a finite symmetric group Sn and specific Young tableau λ corresponding to a numbered partition of n, and consider the action of Sn given by permuting the boxes of λ. Define two permutation subgroups Pλ and Qλ of Sn as follows:[clarification needed]

Pλ={g∈Sn:g preserves each row of λ}

and

Qλ={g∈Sn:g preserves each column of λ}.

Corresponding to these two subgroups, define two vectors in the group algebra ℂSn as

aλ=∑g∈Pλeg

and

bλ=∑g∈Qλsgn⁡(g)eg

where eg is the unit vector corresponding to g, and sgn⁡(g) is the sign of the permutation. The product

cλ:=aλbλ=∑g∈Pλ,h∈Qλsgn⁡(h)egh

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 ℂSn->End⁡(V⊗n) on V⊗n (i.e. V⊗n is a right ℂSn module).

Given a partition λ of n, so that n=λ1+λ2+⋯+λj, then the image of aλ is

Im⁡(aλ):=V⊗naλ≅Symλ1V⊗Symλ2V⊗⋯⊗SymλjV.

For instance, if n=4, and λ=(2,2), with the canonical Young tableau {{1,2},{3,4}}. Then the corresponding aλ is given by

aλ=eid+e(1,2)+e(3,4)+e(1,2)(3,4).

For any product vector v1,2,3,4:=v1⊗v2⊗v3⊗v4 of V⊗4 we then have

v1,2,3,4aλ=v1,2,3,4+v2,1,3,4+v1,2,4,3+v2,1,4,3=(v1⊗v2+v2⊗v1)⊗(v3⊗v4+v4⊗v3).

Thus the set of all aλv1,2,3,4 clearly spans Sym2V⊗Sym2V and since the v1,2,3,4 span V⊗4 we obtain V⊗4aλ=Sym2V⊗Sym2V, where we wrote informally V⊗4aλ≡Im⁡(aλ).

Notice also how this construction can be reduced to the construction for n=2. Let 𝟙∈End⁡(V⊗2) be the identity operator and S∈End⁡(V⊗2) the swap operator defined by S(v⊗w)=w⊗v, thus 𝟙=eid and S=e(1,2). We have that

eid+e(1,2)=𝟙+S

maps into Sym2V, more precisely

12(𝟙+S)

is the projector onto Sym2V. Then

14aλ=14(eid+e(1,2)+e(3,4)+e(1,2)(3,4))=14(𝟙⊗𝟙+S⊗𝟙+𝟙⊗S+S⊗S)=12(𝟙+S)⊗12(𝟙+S)

which is the projector onto Sym2V⊗Sym2V.

The image of bλ is

Im⁡(bλ)≅⋀μ1V⊗⋀μ2V⊗⋯⊗⋀μkV

where μ is the conjugate partition to λ. Here, SymiV and ⋀jV are the symmetric and alternating tensor product spaces.

The image ℂSncλ of cλ=aλ⋅bλ in ℂSn is an irreducible representation of Sn, called a Specht module. We write

Im⁡(cλ)=Vλ

for the irreducible representation.

Some scalar multiple of cλ is idempotent,[1] that is cλ2=αλcλ for some rational number αλ∈ℚ. Specifically, one finds αλ=n!/dim⁡Vλ. In particular, this implies that representations of the symmetric group can be defined over the rational numbers; that is, over the rational group algebra ℚSn.

Consider, for example, S3 and the partition (2,1). Then one has

c(2,1)=e123+e213−e321−e312.

If V is a complex vector space, then the images of cλ on spaces V⊗d provides essentially all the finite-dimensional irreducible representations of GL(V).

## See also

* Representation theory of the symmetric group

## Notes

1. ↑ See (Fulton Harris)

## References

* William Fulton. Young Tableaux, with Applications to Representation Theory and Geometry. Cambridge University Press, 1997.
* Lecture 4 of Fulton, William; Harris, Joe (1991) (in en-gb). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. OCLC 246650103. https://link.springer.com/10.1007/978-1-4612-0979-9.
* Bruce E. Sagan. The Symmetric Group. Springer, 2001.

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