# Spin groups in terms of matrices and/or linear operators

Thus far, the books and articles I have read dealing with spin groups $$\mathbf{Spin}(n)$$ and $$\mathbf{Spin}(p,q)$$ consider them only in terms of either Clifford algebras or topologically as the double covers, respectively, of the special orthogonal groups $$\mathbf{SO}(n)$$ and $$\mathbf{SO}(p,q)$$. The spin groups are Lie groups, and I'm aware that there are exceptional isomorphisms between some of them and a few matrix groups. But my question is in general, can $$\mathbf{Spin}(n)$$ and $$\mathbf{Spin}(p,q)$$ be viewed just in terms of matrices and/or linear operators? If so, what are some good references that might be able to help in answering this question?

• The right thing to google is "spin representation", which provides a natural way to view the spin groups as matrix groups. There are a number of ways to construct the spin representations, but normally one views $Spin(n)$ as a subgroup of the Clifford algebra on $n$ generators and constructs a representation of the Clifford algebra on an exterior algebra. Studying these representations carefully yields the exceptional isomorphisms that you alluded to. The wikipedia page on the spin representation has a lot of detail; otherwise check out Lawson-Michelsohn. – Paul Siegel Nov 11 '18 at 18:29