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.)

  • $\begingroup$ You correctly described the weight lattice; the root lattice is the intersection with that hyperplane rather than the projection. Unfortunately, that's the only part of your question I have much intelligent to say about. $\endgroup$
    – Ben Webster
    Dec 11, 2011 at 14:43
  • $\begingroup$ It seems your pessimism is justified. If theta is a rational multiple of $\pi$, then $2\cos(\theta)$ is an algebraic integer. But if $\theta = \arctan(\sqrt{3}/5)$, then $2\cos(\theta) = 5/\sqrt{7}$. $\endgroup$ Dec 11, 2011 at 20:19
  • $\begingroup$ Have you looked into "scissors congruence"? $\endgroup$ Dec 11, 2011 at 20:32
  • $\begingroup$ @Bruce: Could you explain the reason 'scissors congruence' might be related? Or is it just a hunch? $\endgroup$ Dec 11, 2011 at 20:44
  • $\begingroup$ @Anonymous: Well, that seems to answer the first question. I'm sure that "If theta is a rational multiple of π, then 2cos(θ) is an algebraic integer" is well-known, but it is not well-known to me — can you point me to somewhere to read more? $\endgroup$ Dec 11, 2011 at 22:27

1 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".


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.