Which 4D point groups force an isotropic moment of inertia?

Question

Any three-dimensional object will automatically have an isotropic moment of inertia if it has at least chiral tetrahedral symmetry. What is the situation in four dimensions?

Answer 1

If the object does not have an isotropic moment of inertia, then its symmetry group must be reducible in the language of group representations, i.e., a direct sum of lower-dimensional representations. That is, it must be a subgroup of O(1)×O(3) or O(2)×O(2), or even of O(1)×O(1)×O(2) or O(1)×O(1)×O(1)×O(1). This is easy to see by looking at the ellipsoid E of inertia: Any symmetry must map an axis (or "axial subspace") of E to an axis of the same length.

The finite subgroups of O(4) have been classified, see Classification of finite groups of isometries. Most recently, they are listed in Tables 4.1–4.3 in §4 of the book [CS] by Conway and Smith on quaternions and octonions. I am referring to these tables, which are also reprinted in Chapter 26 of the book "The Symmetries of Things" by Conway, Burgiel, and Goodman-Strauss.

Among the subgroups of O(1)×O(3) (the axial groups), the polyhedral axial groups (the ones that are not subgroups of O(1)×O(1)×O(2)), can be recognized by their "Coxeter names" in Tables 4.2 and 4.3: They are those groups whose Coxeter names have only two numbers between the brackets, like $[3,4]$ or $2.[3,4]$ or $2.[3,3]^+$, or where one of the the three numbers in the brackets is a 2, like $[2,3,3]^+$. There are 21 such groups: 7 chiral ones and 14 achiral ones.

The finite subgroups of O(2)×O(2) are contained the symmetries of duoprisms. (A duoprism is the 4-dimensional Cartesian product of two regular polygons.) They are subgroups of $D_m\times D_n$, where $D_n$ denotes a dihedral group in two dimensions, and $\times$ is used for direct sum of groups. (This is different from the meaning of $D_n$ and $\times$ in the [CS] notation!) In Tables 4.2–4.3, they appear among the groups in the last eight rows (those that don't have Coxeter names) for certain choices of parameters. Some of these groups are subgroups of O(2)×O(2) and thus reducible, but some of them are irreducible but imprimitive: There is decomposition into two orthogonal subspaces $\mathbb R^2\times\mathbb R^2$ which is preserved as a whole (as a decomposition), but where the two subspaces can be swapped by the group. It is not straightforward to distinguish these two types of groups in the [CS] classification. I believe that Tables 4.1 of [CS] contains none of the irreducible groups. (They would have to appear among the last five lines.)

It might be interesting to ask: What are the minimal groups that force an isotropic moment of inertia? In 3D, there is one such group, the rotation group of the tetrahedron, because it is contained in all other polyhedral groups. In 4D, there will be several, including some infinite families.