Which are the point groups where the tensor square of a two dimensional irreducible representation does not decompose into a sum that contains a two-dimensional irreducible representation?

For example (in Schönflies notation) the point group $C_{3v}$ contains a two dimensional representation that is written $E$ in Mulliken symbols for which we have $$ E\otimes E = A_1 \oplus A_2 \oplus E $$ while in the point group $C_{4v}$ the square of $E$ decomposes into $$ E\otimes E = A_1 \oplus A_2 \oplus B_1 \oplus B_2 $$ with all 1-dimensional irreducible representations $A_i$ and $B_i$ (see again Mulliken symbols for the definition a these symbols).

In which point groups in general do we have the former and in which the latter?

The problem arises in the context of the symmetry classification of solutions on a many-body Schrödinger equation. I have noticed that the second case where the product does not decompose into a sum containing a two-dimensional irreducible representation occurs only in point groups that contain only low order generator elements, but I cant make out any deeper reason for that.


  1. A point group is a group together with its representation in $O(3)$.
  2. I mainly am interested here about representations over $\Bbb R$. When representing the point groups over the algebraically closed field $\Bbb C$ (as it 'should' be done) the question is almost unaffected (up to a finite number of exceptions where certain two dimensional irreducible representations over $\Bbb R$ are split into pairs of complex conjugate 1-dimensional irreducible representations over $\Bbb C$). Answers on the latter formulation are equally welcome.
  3. Despite giving non elegant mathematical descriptions (great orthogonalilty theoren looks a bit ugly then) representations over $\Bbb R$ are of interest if for example symmetries of self-adjoint operators are regarded. Then its leads to certain simplifications.
  4. This question could be related, I fail, however to see it clearly.

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