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 $\operatorname{Spin}(n,\mathbf{R}))$ 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:

$$ \operatorname{Z}(\operatorname{Spin}(n,\mathbf{R})) =\operatorname{Z}(\operatorname{Spin}(n))= \begin{cases} \mathrm{Z}_2, n = 2k+1\\ \mathrm{Z}_4, n = 4k+2\\ \mathrm{Z}_2 \oplus \mathrm{Z}_2, n = 4k \end{cases} $$

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?