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Looking at the character table for $\overline{O}$ (double group of $O$) in a book, I noticed that two out of three of the additional irreps (with respect to the five irreps from $O$ itself) are actually just representations of the rotational group ($D^{(j)}$ with j = 1/2 and j = 3/2) restricted to $\overline{O}$ ($D^{(j)}|_{\overline{O}}$). The third irrep than follows from the reduction of $D^{(5/2)}|_{\overline{O}}$.

It is thus possible to setup the character table of $\overline{O}$ by using the table of $O$ for the single-valued irreps and by reducing the appropriate restricted irreps of the rotational group.

My question is:

Is it possible to find all the characters of all the double-valued irreps of a double group $\overline{G}$ just by reducing the $D^{(j)}|_{\overline{G}}$?

And related: Is $D^{(1/2)}|_{\overline{G}}$ always an irrep of the double group?

Edit:

The character table of $\overline{O}$ can be found on page 347 of Group Theory - Application to the Physics of Condensed Matter by M.S. Dresselhaus, G. Dresselhaus and A. Jorio. ISBN 978-3-540-32897-1

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