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 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 "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:

:$P\_lambda=\{\; gin\; S\_n\; :\; g\; mbox\; \{\; preserves\; each\; row\; of\; \}\; lambda\; \}$

and

:$Q\_lambda=\{\; gin\; S\_n\; :\; g\; mbox\; \{\; preserves\; each\; column\; of\; \}\; lambda\; \}$

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

:$a\_lambda=sum\_\{gin\; P\_lambda\}\; e\_g$

and

:$b\_lambda=sum\_\{gin\; Q\_lambda\}\; sgn(g)\; e\_g$

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 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 $V^\{otimes\; n\}=V\; otimes\; V\; otimes\; ...otimes\; V$ ("n" times). Let "S"_n act on this tensor product space by permuting each index. One then has a natural group algebra representation $mathbb\{C\}S\_n\; ightarrow\; mbox\{End\}\; (V^\{otimes\; n\})$ 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 $a\_lambda$ is :$mbox\{Im\}(a\_lambda)\; =\; mbox\{Sym\}^\{lambda\_1\};\; V\; otimes\; mbox\{Sym\}^\{lambda\_2\};\; V\; otimes\; ...\; otimesmbox\{Sym\}^\{lambda\_j\};\; V$

The image of $b\_lambda$ is :where μ is the conjugate partition to λ. Here, $mbox\{Sym\}^\{lambda\}\; V$ and are the symmetric and alternating tensor product spaces.

The image of $c\_lambda\; =\; a\_lambda\; cdot\; b\_lambda$ on $mathbb\{C\}S\_n$ is an irreducible representation [See harv|Fulton|Harris|1991|loc=Theorem 4.3, p. 46] 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 $alpha\_lambdainmathbb\{Q\}$. 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, "S"₃ and the partition (2,1). Then one has $c\_\{(2,1)\}\; =\; e\_\{123\}+e\_\{213\}-e\_\{321\}-e\_\{312\}$

... The image of $c\_lambda$ provides all the finite-dimensional irreducible representations of GL(V) ...

ee also

* Representation theory of the symmetric group

Notes

References

* William Fulton. "Young Tableaux, with Applications to Representation Theory and Geometry". Cambridge University Press, 1997.
* Lecture 4 of Fulton-Harris
* Bruce E. Sagan. "The Symmetric Group". Springer, 2001.