For the following groups, I would like to know the given this group G and its representation R such that the center of G acts trivially (i.e. acts nothing) on R.
Let us denote $\operatorname{Spin}(n,\mathbf{C}))$$\operatorname{Spin}(n,\mathbf{R}))$ as Spin($n$), given the G below:
Spin(3) = Sp(1) = SU(2)
Spin(4) = SU(2) × SU(2)
Spin(5) = Sp(2)
Spin(6) = SU(4)
Spin(7)
Spin(8)
Spin (9)
Spin(10)
Here are the information of the centers:
$$ \operatorname{Z}(\operatorname{Spin}(n,\mathbf{C})) =\operatorname{Z}(\operatorname{Spin}(n))= \begin{cases} \mathrm{Z}_2, n = 2k+1\\ \mathrm{Z}_4, n = 4k+2\\ \mathrm{Z}_2 \oplus \mathrm{Z}_2, n = 4k \end{cases} $$$$ \operatorname{Z}(\operatorname{Spin}(n,\mathbf{R})) =\operatorname{Z}(\operatorname{Spin}(n))= \begin{cases} \mathrm{Z}_2, n = 2k+1\\ \mathrm{Z}_4, n = 4k+2\\ \mathrm{Z}_2 \oplus \mathrm{Z}_2, n = 4k \end{cases} $$
What are examples of such R? Such representation R where the center of above G acts trivially on R?
This question is surely basic, but I would like to know a systematic answer and a full answer -- i.e. finding at least some minimal representations.
e.g. For Spin(3), we see that the center acts on
- the 2-dimensional spinor representation nontrivially as the minus sign of identity.
- the 3-dimensional vector representation (SO(3)) trivially as the identity.
What about the other Spin($n$) cases?