Counting unitariy transformations in SU(N)

Question

Hello everybody,

I'm referring to the following article https://arxiv.org/abs/1810.11563 [1], in particular to section 6.

 The goal is to estimate the number of unitary transformations in $SU(N)$, identifying unitaries within balls of radius $\epsilon$. The strategy is to take the total volume of $SU(N)$ (see also https://arxiv.org/abs/math-ph/0210033 [2]), and then dividing for the volume of an $\epsilon$-ball. Here and in the following $N=2^K$, where $K$ is an integer.

The total volume of $SU(N)$ is:
\begin{equation}
\frac{2 \pi^{\frac{(N+2)(N-1)}{2}}}{1! 2! 3!\cdots(N-1)!}
\end{equation}  
Here the author misses a total factor of $\sqrt{N 2^{N-3}}$, see the original article [2], equation 5.13. Despite this, the author then states that the volume of an $\epsilon$-ball of dimension $N^2-1$ is:
\begin{equation}
\frac{\pi^{\frac{N^2-1}{2}}}{\left(\frac{N^2-1}{2}\right)!}
\end{equation}  
I don't get this last formula. First of all, I believe that there would be a factor $\epsilon$ to some power of $N$. Secondly, it seems to me the volume of an $\epsilon$-ball in an Euclidean space off even dimension, while considering that $N=2^K$, $N^2-1$ is odd. Thirdly, in my opinion it would be better to consider the volume of an $\epsilon$-ball in $SU(N)$ (if I didn't misunderstood the formula above).

My questions are:

- Has someone, reading the article [1], reached a better comprehension than mine on the above formulas?
- If no, may you have other references about estimating the number of unitary transformations in $SU(N)$ within a precision $\epsilon$?
- And, finally, has someone a reference for the volume of an $\epsilon$-ball in $SU(N)$? Looking on the net I didn't find anything until now.

Thank you very much!

  [1]: https://arxiv.org/abs/1810.11563
  [2]: https://arxiv.org/abs/math-ph/0210033

Comments

Exact, the factor is missing. N²-1 is used to get the volume of an euclidian n-sphere but it is not exactly the usual \({\displaystyle V_{N²-1-sphere}={\epsilon ^{{N^2}-1}} {\frac {\pi ^{\frac {{N^2}-1}{2}}}{\Gamma \left({\frac {{N^2}+1}{2}}\right)}}} \)... If you don't find more specific, just for the links: 5.13 might be derived from Kirillov orbit theory, (wikipedia) which states, in connection with representation theory, a duality between the quantum L-R coefficients and the classical Horn problem.