Young symmetrizer

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In mathematics, a Young symmetrizer is an element of the group algebra of the symmetric group, constructed in such a way that the image of the 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.

[edit] Definition

Given a finite symmetric group S_n and specific Young tableau λ corresponding to a numbered partition of n, define two permutation subgroups $P λ$ and $Q λ$ of S_n as follows:

$P_\lambda=\{ g\in S_n : g \mbox { preserves each row of } \lambda \}$

and

$Q_\lambda=\{ g\in S_n : g \mbox { preserves each column of } \lambda \}.$

Corresponding to these two subgroups, define two vectors in the group algebra $\mathbb{C}S_n$ as

$a_\lambda=\sum_{g\in P_\lambda} e_g$

and

$b_\lambda=\sum_{g\in Q_\lambda} \sgn(g) e_g$

where $e g$ is the unit vector corresponding to g, and $sgn(g)$ is the signature of the permutation. The product

c λ = a λ b λ

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.)

[edit] Construction

Let V be any vector space over the complex numbers. Consider then the tensor product vector space $V^{\otimes n}=V \otimes V \otimes ...\otimes V$ (n times). Let S_n act on this tensor product space by permuting each index. One then has a natural group algebra representation $\mathbb{C}S_n \rightarrow \mbox{End} (V^{\otimes n})$ on endomorphisms on $V^{\otimes n}$ .

Given a partition λ of n, so that $n = λ 1 + λ 2 + ... + λ j$ , then the image of $a λ$ is

$\mbox{Im}(a_\lambda) = \mbox{Sym}^{\lambda_1}\; V \otimes \mbox{Sym}^{\lambda_2}\; V \otimes ... \otimes \mbox{Sym}^{\lambda_j}\; V.$

The image of $b λ$ is

$\mbox{Im}(b_\lambda) = \bigwedge^{\mu_1} V \otimes \bigwedge^{\mu_2} V \otimes ... \otimes \bigwedge^{\mu_k} V$

where μ is the conjugate partition to λ. Here, $Sym λ V$ and $\bigwedge^{\mu} V$ are the symmetric and alternating tensor product spaces.

The image $\mathbb{C}S_nc_\lambda$ of $c_\lambda = a_\lambda \cdot b_\lambda$ in $\mathbb{C}S_n$ is an irreducible representation^[1] of S_n, 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 $c^2_\lambda = \alpha_\lambda c_\lambda$ for some rational number $\alpha_\lambda\in\mathbb{Q}$ . 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 $\mathbb{Q}S_n$ .

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).

[edit] See also

Representation theory of the symmetric group

[edit] Notes

^ See (Fulton & Harris 1991, Theorem 4.3, p. 46)

[edit] References

William Fulton. Young Tableaux, with Applications to Representation Theory and Geometry. Cambridge University Press, 1997.
Lecture 4 of Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics, 129, New York: Springer-Verlag, MR 1153249, ISBN 978-0-387-97527-6, ISBN 978-0-387-97495-8
Bruce E. Sagan. The Symmetric Group. Springer, 2001.

[0] See (Fulton & Harris 1991, Theorem 4.3, p. 46)

[1]

Young symmetrizer

From Wikipedia, the free encyclopedia

Contents

[edit] Definition

[edit] Construction

[edit] See also

[edit] Notes

[edit] References

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