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?



Two curious asymptotic results for dimensions of type A objects
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 $V\_\lambda$/$n$!. 


