We know that $Spin(n)/\mathbb{Z}_2=SO(n)$. The $SO(n)$ and $Spin(n)$ have the same Lie algebra. When it comes to the representation of $SO(n)$ and $Spin(n)$, does it make any difference?
$Spin(2n)$ has a $2^n$-dimensional reducible spinor representation, and a $2^{n-1}$-dimensional irreducible spinor representation. Does $SO(2n)$ have the same? How about other representations?
$Spin(2n+1)$ has a $2^n$-dimensional irreducible spinor representation (correct me if I was wrong). Does $SO(2n+1)$ have the same? How about other representations?
Since Spin group is a double cover of SO group, how does this global structure being reflected in the case of representation? (if their representations are the same? or differed also by a double cover? perhaps the parameters of Lie group are "doubled" in some way?) Am I correct to say that SO group has integer spin representations, while Spin group has both integer and half-integer spin representations? For example, the SO(3) group has a trivial representation, and other odd-rank dimensional matrix representation: $$ 1,3,5,7,\dots. $$ In contrast, the Spin(3) group has a trivial representation, and other odd and even-rank dimensional matrix representation: $$ 1,2,3,4,5,6,7,\dots. $$ The odd and even-rank dimensional matrix representation is related to what physicists call the integer and half-integer spin representations.
Do $SO(n)$ and $Spin(n)$ both have complex and real representations? How about quaternion representation?
