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Let $V_{\lambda}$ and $W_{\lambda}$ be the irreducible representations of $S(n)$ and $\mathfrak{su}(N,\mathbb{C})$ associated to the partition $\lambda \in \mathbb{Y}$ of size $| \lambda |=n$ and length $l(\lambda) \leq N$. The following limit $$\frac{\dim V_{\lambda}}{n!} = \lim_{N \rightarrow \infty} \frac{\dim W_{\lambda}}{N^{n}}$$ follows immediately from the well known hook (content) formulas $$\dim V_{\lambda} = \prod_{\square \in \lambda} \frac{n!}{h(\square)} \ \ \ \ \dim W_{\lambda} = \prod_{\square \in \lambda} \frac{N + c(\square)}{h(\square)}$$ which can be found in Macdonald's book. Notice that $n! = \dim_{\mathbb{C}} \mathbb{C}[S(n)]$ and $N^n = \dim_{\mathbb{C}} (\mathbb{C}^N)^{\otimes n}$, so what we're seeing is that as $N \rightarrow \infty$, the relative multiplicity of $V_{\lambda}$ in Schur-Weyl duality approaches the relative multiplicity of $V_{\lambda}$ in the regular representation.

Does anyone have a good feeling for why this is true?

Also, let us not forget the Peter-Weyl theorem! If for a compact group $G$ we write $G^{\vee}$ for its set of finite dimensional irreducible representations over $\mathbb{C}$, we have $$L^2(SU(N)) = \widehat{\bigoplus_{\lambda \in SU(N)^{\vee}}} W_{\lambda} \boxtimes W_{\lambda} $$ $$(\mathbb{C}^N)^{\otimes n}=\bigoplus_{\lambda \in SU(N)^{\vee} \cap S(n)^{\vee}} V_{\lambda} \boxtimes W_{\lambda} $$ $$\mathbb{C}[S(n)] = \bigoplus_{\lambda \in S(n)^{\vee}} V_{\lambda} \boxtimes V_{\lambda}$$

The limit we discussed above relating the second to the third line here actually also happens when we pass from the first to the second line: the ``relative multiplicity'' of $W_{\lambda}$ in its regular representation approaches the relative multiplicity of $W_{\lambda}$ in Schur-Weyl duality.

Can anyone give me some intuition for what's going on here + why I might expect such a result?

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Also, something I really want is some representation theoretic place where I might find a direct sum of $W_{\lambda, N} \boxtimes W_{\lambda, M}$ over all $\lambda \in \mathbb{Y}$ with length $l(\lambda) \leq \min (N,M)$ - a sort of ``unbalanced'' Peter-Weyl theorem for $SU(N)$ on the left and $SU(M)$ on the right. Any thoughts? –  Alexander Moll Mar 31 '12 at 7:09
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The latter thing is the polynomial functions on N by M matrices. The magic word is Howe duality. –  Ben Webster Mar 31 '12 at 13:43
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2 Answers 2

up vote 4 down vote accepted

This is an answer to Alexander's combinatorial reformulation of the question in comments to Bruce's answer.

dim $V_\lambda$/$n$! is the chance that you will get a standard Young tableau if you assign the values 1 to $n$ to the boxes of a tableau of shape $\lambda$ according to a random permutation.

dim $W_\lambda/N^n$ is the chance that you will get a semi-standard Young tableau if you assign a value in $[1,N]$ to each box in a tableau of shape $\lambda$.

Think of the second procedure in the following way: first choose a set of $n$ numbers from 1 to $N$ to serve as entries, and then assign them to boxes. If the entries are all different, then the chances that what you get is a semistandard tableau is the same as the chance that you get a standard tableau starting with 1..$n$.

As $N$ tends to infiniity, the chance that you will choose two entries the same becomes vanishingly small, so in the limit, dim $W_\lambda/N^n$ tends to dim $V_\lambda$/$n$!.

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This is a very nice argument Hugh - thank you! –  Alexander Moll Apr 3 '12 at 22:27
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The $q$-analogue of $\dim W_\lambda$ is the specialisation $s_\lambda(q,q^2,\ldots ,q^n)$ and the $q$-analogue of $\dim V_\lambda$ is the principal specialisation $s_\lambda(q,q^2,\ldots )$ which is manifestly given by taking $n\rightarrow\infty$. In fact in Enumerative Combinatorics II by Stanley this is how the hook length formula for $\dim V_\lambda$ is derived.

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Sorry - I'm familiar with this approach to calculating the limit, and should have included it in my original post, where I was aiming for more of a representation theoretic "Why?". Maybe we can ask this question combinatorially: is there a reason why the number of semi standard Young tableaux of shape $\lambda$ with entries from $\{1, \ldots, N\}$ after dividing by $N^n$ is asymptotic to the number of standard Young tableaux with shape $\lambda$ if we divide by $n!$? –  Alexander Moll Apr 1 '12 at 0:16
    
i.e. is there a ``bijective'' proof of the limit above? –  Alexander Moll Apr 1 '12 at 0:16
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