Fully-diverse multiple-antenna signal constellations and fixed-point-free Lie groups

A group of unitary matrices is called fixed-point-free (fpf) if all non-identity elements of the group have no eigenvalues at unity. Such groups are useful in multiple-antenna communications, especially in multiple-antenna differential modulation, since they constitute a fully-diverse constellation. In this note we consider infinite groups and, in particular, their most interesting case, Lie groups. Two such fpf Lie groups are currently widely used in communications: the group of unit modulus scalars, from which various phase modulation schemes, such as QPSK, are derived, and the 2×2 orthogonal designs of Alamouti, on which many two-transmit-antenna schemes are based. In Lie-group-theoretic jargon these are referred to as U(1) and SU(2). A natural question is whether there exist other fpf Lie groups. We answer this question in the negative: U(1) and SU(2) are all there are