On the form of the finite-dimensional projective representations of an infinite abelian group

Backhouse, N. B.

doi:10.1090/S0002-9939-1973-0318393-6

On the form of the finite-dimensional projective representations of an infinite abelian group
HTML articles powered by AMS MathViewer

by N. B. Backhouse PDF

Proc. Amer. Math. Soc. 41 (1973), 294-298 Request permission

Abstract:

If the locally compact abelian group $G$ has a finite-dimensional unitary irreducible projective representation with factor system $\omega$ (i.e. $G$ has an $\omega$-rep), then a subgroup $G(\omega )$ is defined which fulfils three roles. First, the square-root of the index of $G(\omega )$ in $G$ is the dimension of every $\omega$-rep. Secondly, the $\omega$-reps of $G$ can be labelled by the dual group of $G(\omega )$, up to unitary equivalence. Thirdly, the essential projective form of an $\omega$-rep is determined by a unique projective representation of the finite group $G/G(\omega )$.

References

N. B. Backhouse and C. J. Bradley, Projective representations of abelian groups, Proc. Amer. Math. Soc. 36 (1972), 260–266. MR 308329, DOI 10.1090/S0002-9939-1972-0308329-5

Über die Darstellung endlicher Abelscher Gruppen durch Kollineationen

166

Über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen

127

Adam Kleppner, The structure of some induced representations, Duke Math. J. 29 (1962), 555–572. MR 141735
S. B. Conlon, Twisted group algebras and their representations, J. Austral. Math. Soc. 4 (1964), 152–173. MR 0168663
William G. Harter, Algebraic theory of ray representations of finite groups, J. Mathematical Phys. 10 (1969), 739–752. MR 242969, DOI 10.1063/1.1664901
Adam Kleppner, Multipliers on abelian groups, Math. Ann. 158 (1965), 11–34. MR 174656, DOI 10.1007/BF01370393
George W. Mackey, On induced representations of groups, Amer. J. Math. 73 (1951), 576–592. MR 42420, DOI 10.2307/2372309
George W. Mackey, Unitary representations of group extensions. I, Acta Math. 99 (1958), 265–311. MR 98328, DOI 10.1007/BF02392428
N. B. Backhouse and C. J. Bradley, Projective representations of space groups. I. Translation groups, Quart. J. Math. Oxford Ser. (2) 21 (1970), 203–222. MR 260890, DOI 10.1093/qmath/21.2.203

Untersuchungen über die Darstellung der endlichen Gruppe durch gebrochene lineare Substitutionen

132

The theory of projective representation: pure and applied

A. O. Morris, On a generalized Clifford algebra, Quart. J. Math. Oxford Ser. (2) 18 (1967), 7–12. MR 214612, DOI 10.1093/qmath/18.1.7
A. O. Barut, On irreducible representation of a class of algebras and related projective representations, J. Mathematical Phys. 7 (1966), 1908–1910. MR 224341, DOI 10.1063/1.1704842
Alladi Ramakrishnan and I. V. V. Raghavacharayulu, A note on the representation of Dirac groups, Symposia on Theoretical Physics and Mathematics, Vol. 8 (Symposium, Madras, 1967) Plenum, New York, 1968, pp. 25–32. MR 0243779
Richard Brauer and Hermann Weyl, Spinors in $n$ Dimensions, Amer. J. Math. 57 (1935), no. 2, 425–449. MR 1507084, DOI 10.2307/2371218