Given $n$ vectors $v_1, \ldots, v_n$ in $\mathbb{R}^n$ of course we all know at least one measure for their relative configuration: $|v_1 \wedge\ldots \wedge v_n|$. Now suppose one were given $n$ pairs $(e_1,f_1), \ldots, (e_n,f_n)$ in $\mathbb{R}^{2n}$---then what could be a meaningful measure for their relative configuration? Suppose moreover we require this measure to be $geometric$ ie. having fixed, say a full rank lattice $\Gamma$ in $\mathbb{R}^{2n}$ what could be a meaningful measure for the relative configuration of n 2-planes $\pi_1, \ldots, \pi_n$ in $\mathbb{R}^{2n}$?
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I believe the wedge-based measure you mentioned for the $\mathbb{R}^n$ case works here as well. Assuming each of the $e_i$ and $f_i$ are in $\mathbb{R}^{2n}$, then $\pi_i := e_i \wedge f_i$ is a 2-vector (a weighted specification of a 2-plane in $\mathbb{R}^{2n}$). This wedge quantity will be zero iff $e_i$ and $f_i$ are linearly dependent. Wedging together all of the $\pi_i$ will give a $2n$-vector having analogous geometric meaning as the 2-vector. It will zero iff $X := {e_1, \dots, e_n, f_1, \dots, f_n}$ is a linearly dependent set. Then $|e_1 \wedge f_1 \wedge \dots \wedge e_n \wedge f_n|$ is the norm you are looking for. The geometric quantity this measures should be the signed volume of the simplex spanned by the vectors in $X$ (there may be a normalizing factor, something like $n!$, depending on convention). This is certainly a natural measure of such objects, but probably not the only one. |
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This is more of a musing than an answer, but maybe it can be useful. There is a nice way to think about the Grassmannians of 2-planes in $\mathbb{R}^n$; there is an isomorphism between $Gr(2, n)$ and the projectivized light cone in $\mathbb{C}^n$. Here is how it works: start with your 2-plane $P$ in $\mathbb{R}^n$ and pick any two perpendicular vectors $v$, $w$ of equal norm which span $P$. From those two real vectors, form the complex vector $q = v + i w$. Letting $\cdot$ denote the standard (not hermitian!) dot product, the conditions on $v$ and $w$ imply that $q \cdot q = 0$, so $q$ is a point on the light cone in $\mathbb{C}^n$. A different choice of spanning vectors will cause $q$ to be multiplied by some complex factor $z$, so $q$ is well-defined up to complex scalars. From this perspective you are looking for invariants of configurations of $n$ points on the projective light cone in $\mathbb{C}^{2n}$ (perhaps modulo the action of $O(2n)$ or some other group). This can be cranked out more or less mechanically... |
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This is not a terribly informed answer -- I have just become aware of the subject material (mass and comass) and don't fully understand the geometric significance. Hopefully it will be useful though. Take a look at http://www.encyclopediaofmath.org/index.php/Mass_and_co-mass -- it gives a different norm than the one already mentioned, and in fact has an inequality involving the aforementioned norm. Reference: Herbert Federer, Geometric Measure Theory, pg 38. |
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