I want to determine the branching rules from the quaternion group to the group of the fourth roots of $1$. I have produced the character tables for the groups:
$$\text{The Quaternion Group}$$
and
$$\text{The Fourth Roots of }1.$$
I am just self-learning restricted representations, and I haven't come across a problem like this one yet. I have learned that when you restrict a representation, you remove all of the group elements from the group which do not exist in the subgroup. Thus, for the group of the quaternions, we remove the two classes $\{j,-j\}$ and $\{k,-k\}$. Notice, however, that the group of the quaternions then has $3$ classes while the group of the fourth roots of $1$ has $4$ classes. To determine the relationship between the irreducible representations of a group and its subgroup, I've always used the orthogonality of rows in a character table, but I cannot apply this because the orthogonality principle requires that I have an equal number of columns (classes).
Perhaps this character table approach has limitations and cannot be applied in this case? Or perhaps there is a clever method to resolve the issue that I am facing. Thanks for your help.
As aside on notation:
- I do not often see the gamma notation used on the stack exchange. In the textbook to which I refer (Group Theory and Quantum Mechanics by Michael Tinkham) it is common notation to denote the respresentation of a group element $A$ as $\Gamma(A)$ and thus the irreps are written $\Gamma^{(i)}$ for short.
- My classes for the fourth roots on one are not contained within braces, I'm sorry.