Pin group

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Just as the special orthogonal group SO(n) has a double cover — the spinor group, denoted Spin(n) — the orthogonal group O(n) has two nonisomorphic covering groups, denoted Pin₊(n) and Pin₋(n). These are called the pin groups. (The name is a 'joke', blamed on Serre; "son is to spin as on is to pin".) This strange state of affairs is possible because O(n), (unlike SO(n)) is not connected (its two connected components are sets of matrices with determinant +1 and −1 respectively).

While for O(n) and SO(n), a rotation by 2π is equal to the identity, for the pin groups, as well as for Spin(n), a rotation by 2π is not equal to the identity, even though a rotation by 4π is.

For Pin₊(n), if one repeats the same reflection twice, one gets the identity.

For Pin₋(n), if one repeats the same reflection twice, one gets a rotation by 2π.

There are as many as eight different double covers of Spin(p,q), for . Only two of them are taken to be pin groups, namely, those which admit the Clifford algebra as a representation. They are called Pin(p,q) and Pin(q,p) respectively.