What can be said about number-theoretic properties of the solid angle measures of polytopal cones in the weight lattice of sl(n)?

Question

The following question might be elementary — it is too far from my area of expertise to tell. It has shown up in my research in an interesting way, which I will not go into here, but I'm happy to tell you about it in private if you get in touch with me.

To begin, take the Euclidean space $\mathbb R^n$, and the vector $(1,1,\dots,1)$, and its orthogonal $(n-1)$-dimensional hyperplane. Let's call this hyperplane $\mathfrak h$; it is an $(n-1)$-dimensional Euclidean space. There is the orthogonal projection $\mathbb R^n\to\mathfrak h$. Define a lattice $\Lambda$ inside $\mathfrak h$ as the image of the standard $\mathbb Z^n \subseteq \mathbb R^n$ under this projection. Unless I am mistaken, this lattice is the weight lattice of $\mathfrak{sl}(n)$. (It's one of those weight or root or coroot or something lattices, anyway.)

Pick $n-2$ points in $\Lambda$; then there is an $(n-2)$-dimensional hyperplane in $\mathfrak h$ passing through those points and the origin. With one more bit of data (an orientation, say) those points pick out a half-space (say as those $x\in \mathfrak h$ so that a certain determinant is positive). For want of a better term, let me call an integer polytopal cone (the closure of) a region formed by taking intersections and unions of finitely many such half-spaces. (Is there a better name for such an object?)

For any integer polytopal cone $C$, I can measure its solid angle measure (normalized so that the solid angle measure of $\mathfrak h$ is $1$). Namely, let $B$ denote the unit ball in $\mathfrak h$, and $\operatorname{Vol}$ the standard Euclidean volume function; then the solid angle measure of $C$ is $|C| = \operatorname{Vol}(C\cap B) / \operatorname{Vol}(B)$.

I am interested in understanding what types of numbers can be $|C|$. I would love the answer to the following question to be "yes", but I am not optimistic:

If $C$ is an integer polytopal cone, is $|C|$ necessarily rational?

As I say, I am not optimistic. For example, it seems very unlikely that $\arctan(\sqrt{3}/5)/\pi \approx 0.106147808$ is rational. This number already shows up for $\mathfrak{sl}(3)$. <edit> In the comments below, Anonymous has made clear the following. In general, $|C|$ is not rational. For example, $\arctan(\sqrt{3}/5)/(2\pi)$ is the angle measure of an integer polytopal cone for $\mathfrak{sl}(3)$, and it cannot be rational, by the links that Anonymous suggested. </edit>

So the more general questions are:

What are the number-theoretic properties of the solid angle measures $|C|$?

Is there a large class of integer polytopal cones $C$ for which one can assure that $|C| \in \mathbb Q$?

<edit> I'm particularly interested in the second question. I have a construction that uses these angle measures for some integer polytopal cones. I would be happiest if my construction were to stay within $\mathbb Q$. If there is a large, easily tested class of cones $C$ for which $|C| \in \mathbb Q$, then perhaps I can show that my construction stays within this class. </edit>

(Finally, I have no idea how to tag this question, because it is far from my area of expertise. So I've picked a few tags, and welcome suggestions for retagging.)

Answer 1

$S_n$ acts on the set of cones, and preserves the solid angle measure. If you consider the $n!$ rotations of the cone by elements of $S_n$, and except on the union of some hyperplanes, the number of such cones covering an arbitrary point $x$ in $\mathfrak h$ is constant, say $N$, then the solid angle measure of the cone is $N/n!$.

This argument applies to the cones of the Coxeter arrangement (i.e. the cone generated by the fundamental weights $(1,0,\dots,0), (1,1,0,\dots,0),\dots,(1,\dots,1,0)$ and its $S_n$ translates); the $n!$ rotations are disjoint except on their boundaries and cover $\mathfrak h$, so their volumes are $1/n!$.

Less obviously, this argument also applies to the cone generated by a set of simple roots. This is worked out in Graham Denham's short paper "A note on De Concini and Procesi's curious identity".