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Real Projective Representations of SN and AN

Published online by Cambridge University Press:  20 November 2018

John Q. Huang
John Q. Huang*
Affiliation:
The Fields Institute for Research in Mathematical Sciences, 185 Columbia Street West, Waterloo, Ontario, N2L 5Z5
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Abstract

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Three main results are obtained in this paper: one generalizes the Atiyah-Bott-Shapiro periodicity equivalence on the category of real Clifford modules, (Theorem 2.2); another establishes the existence of two algebras for real projective representations of the symmetric group Sn and the alternating group An, (Theorem 3.2) and determines their structure, (Theorem 6.1); the third describes all the real projective representations of Sn and An except for some small numbers n, (Theorem 7.2).

Keywords

20C2520C1516A2420C30
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 46 , Issue 3 , 01 June 1994 , pp. 543 - 573
Copyright
Copyright © Canadian Mathematical Society 1994

References

1. Adams, J. F., Lectures on Lie Groups, W. A. Benjamin, New York, 1969.Google Scholar
2. Asano, K., Uberdie Darstellungen einer endlichen Gruppedurch réelle Kollineationen, Proc. Imp. Acad. Japan, 1933.Google Scholar
3. Atiyah, M. F., R. Bott and A. Shapiro, Clifford Modules, Topology 3, (Supplement 1) (1964), 3–38.Google Scholar
4. Hoffman, P. N., r-Rings and Wreath Product Representations, Lecture Notes in Mathematics 746, Springer-Verlag, Berlin, 1979.Google Scholar
5. Hoffman, P. N., Projective and Multigraded Representations of Monomial and Multisigned Groups.I. Graded Representations of a twisted Product, Canad. J. Math. 45(1993), 295–339.Google Scholar
6. Hoffman, P. N., A Projective Analogue of Schur's Tensor Power Construction, Comm. Algebra 21(1993), 2211—2249.Google Scholar
7. Hoffman, P. N., Generalizing Clifford Module Periodicity to Graded Representations, (1992), preprint.Google Scholar
8. Hoffman, P. N. and Humphreys, J. F., HopfAlgebras and Projective Representations of GI Sn and GlAn, Canad. J. Math. 38(1986), 1380–1458.Google Scholar
9. Hoffman, P. N. and Humphreys, J. F., Real Projective Representations of Finite Groups, Math. Proc. Cambridge Philos. Soc. 107(1990), 27–32.Google Scholar
10. Hoffman, P. N. and Humphreys, J. F., Projective Representations of the Symmetric Groups, Oxford University Press, Oxford, 1992.Google Scholar
11. Huang, J. Q., Ph.D. Thesis, University of Waterloo, 1993, 1–128.Google Scholar
12. James, G. D., The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics 682, Springer-Verlag, Berlin, 1978.Google Scholar
13. Jôzefiak, T., Characters of Projective Representations of Symmetric Groups, Exposition. Math. 7(1989), 193–247.Google Scholar
14. Karoubi, M., K-Theory : An Introduction, Springer-Verlag, Berlin, Heidleberg, New York, 1977.Google Scholar
15. Hilton, P. J. and U. Stammbach, A Course in Homological Algebra, Graduate Text in Math. 4, Springer-Verlag, New York, 1971.Google Scholar
16. Makhool, M. K. and A. O. Morris, Real Projective Representations of Clifford Algebras, and Symmetric Groups, J. London Math. Soc. (2) 43(1991), 412–420.Google Scholar
17. Nazarov, M. L., An Orthogonal Basis of Irreducible Projective Representations of the Symmetric Groups, Functional Anal. Appl. 22(1988), 77–78.Google Scholar
18. Nazarov, M. L., Young's Orthogonal Form of Irreducible Projective Representations of the Symmetric Groups, J. London Math. Soc. (2) 42(1991), 437–451.Google Scholar
19. Schur, I., Uber die Darstellung der symmetrischen und der alternierenden Gruppen durch gebrochene lineare Substitutionen, J. Reine Angew. Math. 139(1911), 155–250.Google Scholar
20. Schur, I., Uber die reellen Kollineationsgruppen die der symmetrischen oder der alterniernden Gruppe isomorph sind, J. Reine Angew. Math. 158(1927), 63–79.Google Scholar
21. Serre, J. P., Linear Representations of Finite Groups, Graduate Text in Math. 42, Springer-Verlag, New York, 1977.Google Scholar
22. Stembridge, J. R., Shifted Tableaux and the Projective Representations of Symmetric Groups, Adv. in Math. 74(1989), 87–134.Google Scholar