young symmetrizer

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.

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_\lambda

and

Q_\lambda

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

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_\lambda = a_\lambda b_\lambda

is the Young symmetrizer corrseponding 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.

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=\lambda_1+\lambda_2+ ... +\lambda_j

, then the image of

a_\lambda

\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_\lambda

\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,

\mbox{Sym}^{\lambda} V

and

\Lambda^{\mu} V

are the symmetric and alternating tensor product spaces. The image of

c_\lambda = a_\lambda \cdot b_\lambda

is then an irreducible representation of S_n. We write

\mbox{Im}(c_\lambda) = V_\lambda

for the irreducible representation. Note that some scalar multiple of

c_\lambda

is idempotent, that is

c^2_\lambda = \alpha_\lambda c_\lambda

for some rational number

\alpha_\lambda\in\mathbb{Q}

. Specifically, one finds

\alpha_\lambda=n! / \mbox{dim } V_\lambda

. 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

c_\lambda

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

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 0-387-974495-4 See Chapter 4
Bruce E. Sagan. The Symmetric Group. Springer, 2001.

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