In string theory, an important role is played by certain Lie groups which are locally isomorphic to SO(4m),m⩽8. It has long been known that these groups are actually isomorphic not to SO(4m) but rather to the groups for which the half-spin representations are faithful, which we propose to call Semispin(4m). (They are known in the physics literature by the ambiguous name of “Spin(4m)/Z2.”) Recent work on string duality has shown that the distinction between SO(4m) and Semispin(4m) can have a definite physical significance. This work is a survey of the relevant properties of Semispin(4m) and its subgroups.

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