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.

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.

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,
• 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.

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.