Consider the following matrix
$$\Gamma(4) = \begin{pmatrix} 0 & 1 & 0 & 0\\ -1 & 0 & \sqrt{2} & 0\\ 0 & -\sqrt{2} & 0 & \sqrt{3}\\ 0 & 0 & -\sqrt{3} &0 \end{pmatrix}$$
So it's easy to understand the form of $\Gamma(n)$: it is skew-symmetric, with two diagonal lines with entries {$\pm\sqrt{1}, \pm\sqrt{2},\dots, \pm\sqrt{n}$}. My question is how to get the large $n$ behavior for the summation of powers of its singular values:
$$\sum_i \sigma_i ^p$$
The above expression is also related to Schatten p-norm. For example, it's not difficult to show that
$$ \sum_i \sigma_i^2 = \|\Gamma(n)\|_2^2 \sim n^2/2 $$
However, I also need to know the large $n$ behaviour for $p=1,4$. I wrote a small code to do a fitting which suggests that
$$ \sum_i \sigma_i \sim n^{3/2} $$
and
$$ \sum_i \sigma_i^4 \sim n^{3} $$
Any idea how to prove them? It seems like the asympotic behavior is the same as the summation of the matrix entries (see What is $\sum\limits_{i=1}^n \sqrt i\ $?)
$$\sum_i (\sqrt{i})^p$$
