I am looking for a quick reference for the volume formula for all the compact simple Lie groups embedded as matrix groups in the natural way. The one I care most for are the real orthogonal groups. I don't see how to deduce them from the Weyl integration formula. I believe they are of the order diameter raised to the power of the dimension. So for instance for $SO(n)$ the volume should be of order $n^{\Theta(n^2)}$.
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7$\begingroup$ For the orthogonal group $SO(n)$, isn't it just the product of the volumes of the $k$-spheres in their natural embeddings, as $k$ ranges from 1 to $n{-}1$? Similarly, for $U(n)$, it should be the product of the $2k{-}1$-spheres as $k$ ranges from $1$ to $n$, and so on. $\endgroup$– Robert BryantCommented Jan 4, 2012 at 1:26
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$\begingroup$ Oh you are right. Thanks for the observation! $\endgroup$– John JiangCommented Jan 4, 2012 at 1:29
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4$\begingroup$ This is not quite true: $SO(2)$ as a circle in the vector space of $2\times 2$ matrices has radius $\sqrt 2$, in the metric that I am thinking of. $\endgroup$– Tom GoodwillieCommented Jan 4, 2012 at 1:38
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4$\begingroup$ @Tom: You are quite right. I forgot about the factor of $(\sqrt{2})^{k-1}$ that you have to put in at each level because the natural map $\pi:SO(k)\to S^{k-1}$ is not a Riemannian submersion, but is a Riemannian submersion scaled by a factor of $1/\sqrt{2}$, i.e., each horizontal vector for this bundle is shrunk by a factor of $\sqrt{2}$ by the differential of $\pi$. Thus, the overall factor you need to multiply the answer I gave by is $(\sqrt{2})^{n(n-1)/2}$. The recipe I gave for $U(n)$ is not right either, as the map $\pi:U(k)\to S^{2k-1}$ shrinks horizontal volumes by $2^{k-1}$. Sorry. $\endgroup$– Robert BryantCommented Jan 4, 2012 at 2:21
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1$\begingroup$ @J A S O N: I think you can see it from the Lie algebra $so(n)$, which spans an $n(n-1)/2$ dimensional subspace of $R^{n^2}$. The differential of the submersion corresponds to the projection of $so(n)$ to the first column vector space. Vertical vectors of the said bundle are thus those vectors in $so(n)$ that get mapped to 0, and horizontal ones are orthogonal to the vertical ones, so must be of the form $e_i \wedge e_1$. Now these have length $\sqrt{2}$ each under Euclidean metric of the ambient space. But they get mapped to $e_i$ by $d\pi$, which have length $1$ each! $\endgroup$– John JiangCommented Jan 10, 2017 at 4:18
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3 Answers
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As far as I know, the most standard volume formula was obtained by I.G. Macdonald in a very short 1980 paper here. Since the volume depends on the choice of Haar measure, Macdonald starts with the (complexified) Lie algebra of the compact Lie group, fixing a Lebesgue measure there along with a fixed lattice such as a Chevalley $\mathbb{Z}$-form. The second ingredient in his formula comes from the standard invariants in the cohomology calculation for the group.
There is another approach to Macdonald's formula in a later paper by Y. Hashimoto: On Macdonald’s formula for the volume of a compact Lie group., Comment. Math. Helv. 72 (1997), no. 4, 660–662.
answered Aug 12, 2013 at 10:54
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Explicit formulas for the volumes of compact Lie groups, with respect to their Haar measures, given in terms of their root data are given in: M. S. Marinov: Invariant volumes of J. Phys A: Math. Gen. 13 (1980) 3357-3376. Availabe in Prof. Marinov's memorial site.
The final formula is given in equation 19, which is also tabulated for the various simple types. The article compares the general formula with the sphere based computations for some of the classical groups.
The method of computation in the article is based on the Weyl's integration formula, however, the volume of the flag manifold (called the orbit space in the article) was read from the evolution operator of the Laplacian on the group manifold. Prof. Marinov's interest in this subject was due to his work in quantum mechanics and path integrals on group manifolds.
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$\begingroup$ I have actually run into this article before but the formula is a bit hard for me to parse. My interest comes from analyzing the Kac random walk on $SO(n)$. $\endgroup$ Commented Jan 4, 2012 at 8:57
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The volume of $O(n)$ is worked out in Muirhead's book `Aspects of multivariate statistical theory' (Wiley, 1982). See Corollary 2.1.16.
