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.

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[edit] Definition

Given a finite symmetric group Sn and specific Young tableau λ corresponding to a numbered partition of n, define two permutation subgroups Pλ and Qλ of Sn 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 eg is the unit vector corresponding to g, and sgn(g) is the signature of the permutation. The product

cλ = aλbλ

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 Sn 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 = λ1 + λ2 + ... + λj, then the image of aλ 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 bλ is

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

where μ is the conjugate partition to λ. Here, SymλV and ΛμV are the symmetric and alternating tensor product spaces.

The image of c_\lambda = a_\lambda \cdot b_\lambda is then an irreducible representation of Sn. We write

Im(cλ) = Vλ

for the irreducible representation.

Note that some scalar multiple of cλ is idempotent, that is c^2_\lambda = \alpha_\lambda c_\lambda for some rational number \alpha_\lambda\in\mathbb{Q}. Specifically, one finds αλ = n! / dim Vλ. In particular, this implies that representations of the symmetric group can be given in terms of 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) = ...

... The image of cλ provides all the finite-dimensional irreducible representations of GL(V) ...

[edit] See also

[edit] References

  • William Fulton. Young Tableaux, with Applications to Representation Theory and Geometry. Cambridge University Press, 1997.
  • William Fulton and Joe Harris, Representation Theory, A First Course (1991) Springer Verlag New York, ISBN 0-387-97495-4 See Chapter 4
  • Bruce E. Sagan. The Symmetric Group. Springer, 2001.
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