Invariants of a set of real unit vectors in 3d space, under SO(3)

2012-11-14T15:49:44Z

I have a set of $n$ real unit vectors, in 3-dimensional space.

(It is a follow up of Sets of vectors related by a rotation.)

Is there a construction providing a complete set of independent*) invariants under SO(3)?

*) I'm the most interested in independent invariants, but a solution without this assumption is of mine interest as well.

I had two initial ideas, but I run in some problems:

construct Grammian matrix of vectors; however, then the matrix itself depend on a particular permutation of vectors, making it impossible to use as if two set of vectors are equivalent; also, determinant of the matrix is not enough as for $n=3$ it gives only 2 out of 3 invariants,
calculate moments, i.e. $M_{ijk} = \langle x^i y^j z^k \rangle$; then for $i+j+k=1$ there is length squared, for $i+j+k=2$ characteristic polynomial of the inertia matrix, but I don't know how to go further.

Answer by Robert Bryant for Invariants of a set of real unit vectors in 3d space, under SO(3)

2012-11-15T22:44:19Z

The following construction reduces your problem it to a classical and well-studied problem in invariant theory. First, I claim that there is a natural way to interpret an $n$-tuple of points in $S^2$ as a point of $\mathbb{CP}^n$ and vice-versa. This depends on interpreting $S^2$ as $\mathbb{CP}^1$ and the classical fact that the symmetric product of $n$ copies of $\mathbb{CP}^1$ is naturally regarded as $\mathbb{CP}^n$. Moreover, this can be done in an $\mathrm{SO}(3)$-equivariant way, so that your problem becomes that of finding invariants for an action of $\mathrm{SO}(3)$ on $\mathbb{CP}^n$, a very classical problem.

To see this, remember that $\mathrm{SU}(2)$, which is the double cover of $\mathrm{SO}(3)$ acts on $\mathbb{C}^2$ in the obvious way and that this action is transitive on the $1$-dimensional subspaces $L\subset\mathbb{C}^2$, the set of which is naturally $S^2=\mathbb{CP}^1$. Moreover, the induced action on $S^2$ is that of $\mathrm{SO(3)}=\mathrm{SU}(2)/\{\pm I_2\}$. Thus, nonzero vectors in $\mathbb{C}^2$ up to nonzero complex multiples correspond to points of $S^2$, and, given any $n$-tuple of unit vectors $u_i\in S^2$ for $1\le i\le n$, they can be represented by an $n$-tuple of lines in $\mathbb{C}^2$ of the form $u_i = [v_i] = \mathbb{C}\cdot v_i\subset \mathbb{C}^2$. Now consider the symmetric product $$ v_1v_2\cdots v_n\in \mathsf{S}^n(\mathbb{C}^2) \simeq \mathbb{C}^{n+1} $$ The line spanned by this product, $[v_1v_2\cdots v_n]\in \mathbb{CP}^n$ is well-defined, independent of the choice of the $v_i$ to represent the $u_i$. Conversely, since any nonzero complex polynomial in two variables that is homogeneous of degree $n$ can be factored into linear factors uniquely up to complex multiples and permutations, it follows that any element $p\in \mathbb{CP}^n$ can be constructed this way and, moreover, the $n$-tuple of points $u_i\in S^2 = \mathbb{CP}^1$ that gives rise to $p$ is uniquely determined up to permutation.

Now, what about the action of $\mathrm{SO}(3)$? Since this is induced by the action of $\mathrm{SU}(2)$ on $\mathbb{C}^2$, and that action extends in the usual way to the symmetric power $\mathsf{S}^2(\mathbb{C}^2)=\mathbb{C}^{n+1}$, on which it is irreducible (as a complex representation), it follows that this construction is $\mathrm{SO}(3)$-equivariant.

Thus, you are reduced to finding a 'complete' (in your sense) and 'independent' (in your sense) set of invariants for the action of $\mathrm{SO}(3)$ on $\mathbb{CP}^n$ that is induced by the irreducible representation of $\mathrm{SU}(2)$ on $\mathbb{C}^{n+1}$. This is, of course, a very classical problem, about which an enormous amount has been known since the 19th century. In particular, the Clebsch-Gordan formulae can be used to give one a procedure for generating all of the polynomial invariants, but they inevitably have complicated relations among them as soon as $n$ gets bigger than $3$ or $4$.

Invariants of a set of real unit vectors in 3d space, under SO(3) - MathOverflow

Invariants of a set of real unit vectors in 3d space, under SO(3)

Answer by Robert Bryant for Invariants of a set of real unit vectors in 3d space, under SO(3)