My question concerns how to write the center of the Spin group written in the matrix of Spinor representation.
(1). For example, for $Spin(3)=SU(2)$, we have that the center
$$Z(Spin(3))=Z(SU(2))=\mathbb{Z}/2.$$
If we express the $Spin(3)=SU(2)$ in terms of the spinor representation (the 2-dimensional representation), we write the $Spin(3)=SU(2)$ as a rank-2 matrix.
Question: Now how can we write the center of the $Spin(3)$ group written in the matrix of Spinor representation as a rank-2 matrix:
Ans: $$ \begin{pmatrix} -1& 0 \\ 0& -1 \end{pmatrix}. $$
(2). Now, my question, for $Spin(6)=SU(4)$, we have that the center
$$Z(Spin(6))=\mathbb{Z}/4.$$
If we express the $Spin(6)$ in terms of the spinor representation (the $2^3=8$-dimensional representation), we write the $Spin(6)$ as a rank-8 matrix.
Question: How can we write the center of the $Spin(6)$ group written in the matrix of Spinor representation as a rank-8 matrix?
(3). Now, my question, for $Spin(10)$, we have that the center
$$Z(Spin(10))=\mathbb{Z}/4.$$
If we express the $Spin(10)$ in terms of the spinor representation (the $2^5=32$-dimensional representation), we write the $Spin(10)$ as a rank-32 matrix.
Question: How can we write the center of the $Spin(10)$ group written in the matrix of Spinor representation as a rank-32 matrix or a rank-16 matrix?
p.s. Helpful Refs on Spinor representation.