Combinatorial Representation Theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 17 September 2013

Appendix A

A5. Schur-Weyl duality and Young symmetrizers

Let $V$ be the usual $n\text{-dimensional}$ representation of $GL(n,ℂ)$ on column vectors of length $n,$ that is $$V=\text{span}\{b_1,\dots ,b_n\}\text{where}{b}_i={(0,\dots ,0,1,0,\dots ,0)}^t,$$ and the $1$ in $b_i$ appears in the $i\text{th}$ entry. Then $$V^{\otimes k}=\text{span}\{b_{i_1}\otimes \cdots \otimes b_{i_k}\hspace{0.17em}|\hspace{0.17em}1\le i_1,\dots ,i_k\le n\}$$ is the span of the words of length $k$ in the letters $b_i$ (except that the letters are separated by tensor symbols). The general linear group $GL(n,ℂ)$ and the symmetric group $S_k$ acton $V^{\otimes k}$ by $$g(v_1\otimes \cdots \otimes v_k)=g{v}_1\otimes \cdots \otimes g{v}_k,\text{and}(v_1\otimes \cdots \otimes v_k)\sigma =v_{\sigma \left(1\right)}\otimes \cdots \otimes v_{\sigma \left(k\right)},$$ where $g\in GL(n,ℂ),$ $\sigma \in S_k,$ and $v_1,\dots ,v_k\in V\text{.}$ (We have chosen to make the $S_k\text{-action}$ a right action here, one could equally well choose the action of $S_k$ to be a left action but then the formula would be $\sigma (v_1\otimes \cdots \otimes v_k)=v_{{\sigma }^{-1}\left(1\right)}\otimes \cdots \otimes v_{{\sigma }^{-1}\left(k\right)}\text{.)}$ The following theorem is the amazing relationship between the group $S_k$ and the group $GL(n,ℂ)$ which was discovered by Schur [Sch1901] and exploited with such success by Weyl [Wey1946].

Theorem A5.1. (Schur-Weyl duality)

(a)	The action of $S_k$ on $V^{\otimes k}$ generates ${\text{End}}_{GL(n,ℂ)}\left(V^{\otimes k}\right)\text{.}$
(b)	The action of $GL(n,ℂ)$ on $V^{\otimes k}$ generates ${\text{End}}_{S_k}\left(V^{\otimes k}\right)\text{.}$

This theorem has the following important corollary, which provides a intimate correspondence between the representation theory of $S_k$ and some of the representations of $GL(n,ℂ)$ (the ones indexed by partitions of $k\text{).}$

Corollary A5.2. As $GL(n,ℂ)\times S_k$ bimodules $$V^{\otimes k}\cong \underset{\lambda ⊢k}{⨁}{V}^{\lambda }\otimes S^{\lambda },$$ where $V^{\lambda }$ is the irreducible $GL(n,ℂ)\text{-module}$ and $S^{\lambda }$ is the irreducible $S_k\text{-module}$ indexed by $\lambda \text{.}$

If $\lambda$ is a partition of $k,$ then the irreducible $GL(n,ℂ)\text{-representation}$ $V^{\lambda }$λ is given by $$V^{\lambda }\cong V^{\otimes k}P\left(T\right)N\left(T\right),$$ where $T$ is a tableau of shape $\lambda$ and $P\left(T\right)$ and $N\left(T\right)$ are as defined in Section 2, Question C.

Notes and references

This is the survey paper Combinatorial Representation Theory, written by Hélène Barcelo and Arun Ram.

Key words and phrases. Algebraic combinatorics, representations.

Barcelo was supported in part by National Science Foundation grant DMS-9510655.
Ram was supported in part by National Science Foundation grant DMS-9622985.
This paper was written while both authors were in residence at MSRI. We are grateful for the hospitality and financial support of MSRI..

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