# Representation R where the center of Spin group acts trivially on R

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?

• I am confused by your second sentence. By $\operatorname{Spin}(n, \mathbb{C})$, do you mean a double cover of $SO(n, \mathbb{C})$? This is not the same as $\operatorname{Spin}(n)$. – Michael Albanese Dec 18 '19 at 23:38
• Should be $\operatorname{Spin}(n,\mathbf{R})$. But I follow the Wikipedia for the center group – annie marie heart Dec 19 '19 at 2:53
• en.wikipedia.org/wiki/Spin_group#Center – annie marie heart Dec 19 '19 at 2:54
• Presumably in the odd case the centre is the kernel of the projection to SO(n) ? Then the representations trivial on the centre are the representations of SO(n). Maybe for the others you can see this from the representation ring of Spin(n) by direct calculation of the action of the centre ? You should be able to find the representation ring of Spin(n) in lots of places. For example Husemoller's Fibre Bundles I know describes it but otherwise a book on representations of compact Lie groups. – Michael Murray Dec 19 '19 at 4:58