## Introduction

Let's fix $m\in \mathbb N$. For each n, the unitary group $\mathbf U(m)$ is represented in the space of tensors of rank $n$ over $\mathbb C^m$ $$V_{n,m}=\bigotimes_{k=1}^n \mathbb C^m$$ and the symmetric group $S_n$ acts on $V_{n,m}$ by permutation of factors. Now the space $V_{n,m}$ breaks into the direct sum of subspaces $V_{n,m}(\lambda)$ which are primary with respect to each of these actions and irreducible with respect to the joint action of $S_n\times \mathbf U(m)$ $$V_{n,m}=\bigoplus_{\lambda \in \mathbb{Y}(n,m)}V_{n,m}(\lambda)$$

Where $\lambda$ ranges over all Young diagrams of size $n$ with at most $m$ rows (So $\lambda \in \mathbb{Y}(n,m)$ means $\lambda=(\lambda_1,\dots ,\lambda_k)$ is a partition of $n$ with $k\le m$). The tensors from $V_{n,m}(\lambda)$ are said to have symmetry type $\lambda$. We define the relative dimensions $$d_{n,m}(\lambda)=\frac{\dim V_{n,m}(\lambda)}{\dim V_{n,m}}$$ which tell us how tensors are distributed into symmetry types.

## Motivation and Question

I was reading Kerov's "Asymptotic representation theory of the symmetric group and it's aplications in analysis" and was trying to provide proofs for some of the results stated there. (He does give references, which I can't reach at the moment.)

The following two theorems are due to Kerov

Theorem 1If for each $\lambda$ we associate $x=(x_1,\dots,x_m)$ with $x_k=\frac{\lambda ^{(n)}_k-n/m}{\sqrt{n}}$. The joint distribution of $x_k$'s as $n\to \infty$ with respect to the measure $d_{n,m}$ on $\mathbb{Y}(n,m)$ weakly converges to an absolutely continuous measure on the cone $C_m=\{x: x_1\geq x_2\geq\cdots \geq x_m \;;\;\sum x_k=0\}$ with density $$\phi _m(x)=c \prod _{i < j}(x_i-x_j)^2 e^{-m/2 \sum x_k^2}$$ where $$c=\frac{m^{(m-1)m/2}}{1!2!\cdots (m-1)!}\left(\frac{m}{2\pi}\right)^{(m-1)/2}$$

.

Theorem 2Let $\lambda ^{(n)}\in \mathbb{Y}(n,m)$ be the Young diagram for which the tensors of type $\lambda ^{(n)}$ are most probable. Then $$\lim_{n\to \infty} \frac{\lambda ^{(n)}_k-n/m}{\sqrt{n/m}}=z_k$$ for each $k=1,2,\dots,m$ where $z_1,z_2,\dots z_m$ are the roots of the Hermite polynomial $H_m(z)$

I can prove Theorem 2 assuming Theorem 1, but I don't see a nice argument for proving the first theorem itself. Can anyone provide a sketch of the proof, or some hint how to approach Theorem 1?