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OK, so I'm reading about this nice measure you can define on a (real) Grassmannian on Wikipedia. Basically, and to save you the trip through the link, consider the Haar measure $\theta$ on $O(n)$, fix a space V in your Grassmannian. Then for any subset A, the measure of A is $$\gamma(A)=\theta(g\in O(n) \mid gV \in A).$$

Fair enough. Two things Wikipedia does not really tell me, though:

  1. Where does this construction originate? I would imagine something like this to be fairly folklore, but it would sure be nice if someone has a reference.

  2. Does anyone know of especially interesting applications of this measure? Since the Wikipedia article cruelly lacks context, I would really like to see this idea in action.

Thanks in advance!

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4 Answers 4

This measure is the unique $O(n)$ invariant measure on the Grassmannian up to a multiplication by a scalar. The following lecture note explains invariant measures on homogeneous spaces. One application is in harmonic analysis on homogeneous spaces, see for example: the following review article

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I am sorry but the first link is invalid. –  Leonid Petrov Aug 15 '10 at 14:22
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@Leonid Petrov: Just remove the trailing period from the linked URL in David Bar Moshe's comment, i.e. www-math.mit.edu/~dav/integration.ps –  gspr Aug 15 '10 at 14:28
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For one application, an explicit curve {C(t) : t in [0,oo)}, dense in a Grassmannian of 2-planes in n-space, is the basis for the animation technique in statistical computer graphics known as the Grand Tour. It's important to ensure that as t -> oo, the curve C spends time in any open set U proportional to the invariant measure* of U. * Though the invariant measure on a Grassmannian is unique up to a scalar multiple, the invariant metric is not in the sole case of 2-planes in 4-space. This oriented Grassmannian's metric is the product of two round 2-spheres whose radii may be in any ratio. –  Daniel Asimov Aug 15 '10 at 15:45
    
Thanks Daniel. This sounds like a rather intriguing application, and I don't imagine I could have stumbled upon it by myself. –  Thierry Zell Aug 15 '10 at 21:37
    
David: your lecture notes have been very enlightening. Thanks! –  Thierry Zell Aug 16 '10 at 13:20

A nice application is a Crofton-like formula for the codimension $k=\dim V$ submanifolds of $S^{n-1}\subset\mathbb{R}^n$. If $X$ is such a submanifold, then its $n-k-1$ dimensional measure is the average number of points of $V\cap X$, $V\in\mathrm{Gr}_{n,k}$, with respect to the said measure, multiplied by half the measure of $S^{n-k-1}$ (which has 2 intersection point for almost all $V$).

There also is an affine version with the grassmannian of $k$-dimensional affine subspaces of (euclidean) $\mathbb{R}^n$ and codimension $k$ submanifolds of $\mathbb{R}^n$ (the original Crofton formula is the case $n=2$, $k=1$).

Appropriate keywords would be integral geometry, and perhaps also geometric measure theory.

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It's easy to describe the metric that gives rise to this measure: define a map from the Grassmannian of $k$-planes in $\mathbb{R}^n$ to the set of $n$ by $n$ matrices by associating to a $k$-plane $V$ the orthogonal projection $\pi_V$ onto $V$. This embeds the Grassmannian as a real algebraic subvariety of the space of $n$ by $n$ matrices (characterized as the set of symmetric matrices $\pi$ with trace $k$ such that $\pi^2=\pi$) and there is a natural choice of metric given by $d(V,W)=|\pi_V-\pi_W|$ where $|\cdot|$ denotes the sup norm on the space of $n$ by $n$ matrices. It follows from the definition that the metric is $O(n)$-invariant and therefore gives rise to an $O(n)$-invariant measure (up to scalars the one you ask about).

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First of all, a general fact. Any transitive homogeneous space $X$ of a compact group $K$ has a unique $K$-invariant probability measure. Existence: take the image $\nu$ of the normalized Haar measure $m$ on $K$ under the map $g\mapsto gx_0$ (that's the construction you refer to). The measure $\nu$ is well-defined, since the mass of $m$ is finite, it does not depend on $x_0\in X$ by right invariance of $m$, and is $K$-invariant by left invariance of $m$. Uniqueness: take an arbitrary $K$-invariant measure $\nu'$ on $X$, and consider its convolution $m\ast\nu'$ with the measure $m$ (i.e., the image of the product of $m$ and $\nu'$ under the map $(g,x)\mapsto gx$). Then, on one hand $m\ast\nu'=\nu'$ by $K$-invariance of $\nu'$, on the other hand $m\ast\nu'=\nu$ by the above construction.

Thus, since the Grassmannian in question has a transitive compact group of automorphisms, it carries a "natural" invariant measure. So that "platonically" it is always there - like, for instance, the Riemannian volume on a Riemannian manifold (by the way, as mentioned before, any invariant Riemannian metric on the Grassmannian produces the measure in question in this way).

However, there is one subtlety here which has so far remained unnoticed. In order to define the Grassmannian one needs a linear structure, whereas the orthogonal group $O(n)$ is defined in terms of the Euclidean structure. Therefore, the "canonical" measure we are talking about is only canonical with respect to the given Euclidean structure on the linear space $V=R^n$. So, if we look at the problem from this point of view, we obtain a map which assigns to any Euclidean structure on $V$ a probability measure on the Grassmannian $Gr_k(V)$. In fact, this measure depends only on the projective class of the Euclidean structure (i.e., on the corresponding similarity structure), which are parameterized by the Riemannian symmetric space $S=SL(n,R)/SO(n)$ (equivalently, one can say that we consider only the Euclidean structures on $V$ with the same volume form, whence $SL$ instead of $GL$).

Thus, we have a map $x\mapsto\nu_x$ from $S$ to the space $P(Gr_k)$ of probability measures on $Gr_k$. One can show that this map is an injection, so that it can be used in order to compactify the symmetric space $S$ by taking its closure in the weak$^*$ topology of $P(Gr_k)$. This is an example of a so-called Satake-Furstenberg compactification, which can be defined for an arbitrary non-compact Riemannian symmetric space. In the case of the space $S=SL(n,R)/SO(n)$ all such compactifications are obtained by considering rotation invariant measures on the flag space of $V$ and its equivariant quotients (in particular, Grassmannians). In the general case the role of the flag space is played by the so-called Furstenberg boundary, which is the quotient of the semi-simple Lie group by its minimal parabolic subgroup. The most recent reference for all this is the book by Borel and Ji.

The simplest non-compact symmetric space is the hyperbolic plane. In this case the Furstenberg boundary (the associated "flag space") is just the boundary circle in the disk model. Each point of the hyperbolic plane determines a unique probability measure on the boundary circle invariant with respect to the rotations around this point. These measures appear in the classical Poisson formula for bounded harmonic functions in the unit disk (usually it is written in terms of just a single measure corresponding to the Euclidean center of the disk; the other measures appear in the guise of their Radon-Nikodym derivatives with respect to this one, which is precisely the Poisson kernel).

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