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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.


Definition

Given a finite symmetric group Sn and specific Young tableau λ corresponding to a numbered partition of n, define two permutation subgroups <math>P_\lambda</math> and <math>Q_\lambda</math> of Sn as follows:

<math>P_\lambda=\{ g\in S_n : g \mbox { preserves each row of } \lambda \}</math>

and

<math>Q_\lambda=\{ g\in S_n : g \mbox { preserves each column of } \lambda \}</math>

Corresponding to these two subgroups, define two vectors in the group algebra <math>\mathbb{C}S_n</math> as

<math>a_\lambda=\sum_{g\in P_\lambda} e_g</math>

and

<math>b_\lambda=\sum_{g\in Q_\lambda} \sgn(g) e_g</math>

where <math>e_g</math> is the unit vector corresponding to g, and <math>\sgn(g)</math> is the signature of the permutation. The product

<math>c_\lambda = a_\lambda b_\lambda</math>

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 <math>V^{\otimes n}=V \otimes V \otimes …\otimes V</math> (n times). Let Sn act on this tensor product space by permuting each index. One then has a natural group algebra representation <math>\mathbb{C}S_n \rightarrow \mbox{End} (V^{\otimes n})</math> on endomorphisms on <math>V^{\otimes n}</math>.

Given a partition λ of n, so that <math>n=\lambda_1+\lambda_2+ … +\lambda_j</math>, then the image of <math>a_\lambda</math> is

<math>\mbox{Im}(a_\lambda) =

\mbox{Sym}^{\lambda_1}\; V \otimes
\mbox{Sym}^{\lambda_2}\; V \otimes … \otimes
\mbox{Sym}^{\lambda_j}\; V
</math>

The image of <math>b_\lambda</math> is

<math>\mbox{Im}(b_\lambda) =

\bigwedge^{\mu_1} V \otimes
\bigwedge^{\mu_2} V \otimes … \otimes
\bigwedge^{\mu_k} V
</math>
where μ is the conjugate partition to λ. Here, <math>\mbox{Sym}^{\lambda} V </math> and <math>\bigwedge^{\mu} V</math> are the symmetric and alternating tensor product spaces.

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

<math>\mbox{Im}(c_\lambda) = V_\lambda</math>

for the irreducible representation.

Note that some scalar multiple of <math>c_\lambda</math> is idempotent, that is <math>c^2_\lambda = \alpha_\lambda c_\lambda</math> for some rational number <math>\alpha_\lambda\in\mathbb{Q}</math>. Specifically, one finds <math>\alpha_\lambda=n! / \mbox{dim } V_\lambda</math>. 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 <math>\mathbb{Q}S_n</math>.

Consider, for example, S3 and the partition (2,1). Then one has <math>c_{(2,1)} = …</math>

… The image of <math>c_\lambda</math> provides all the finite-dimensional irreducible representations of GL(V) …


See also

  • Representation theory of the symmetric group


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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