Real Projective Representations of SN and AN

Huang, John Q.

Canadian journal of mathematics, Tome 46 (1994) no. 3, pp. 543-573

Résumé

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).

Détail

DOI : 10.4153/CJM-1994-029-9

Mots-clés : 20C25, 20C15, 16A24, 20C30

Bibliographie

[1] 1. Adams, J. F., Lectures on Lie Groups, W. A. Benjamin, New York, 1969. Google Scholar

[2] 2. Asano, K., Uberdie Darstellungen einer endlichen Gruppedurch réelle Kollineationen, Proc. Imp. Acad. Japan, 1933. Google Scholar

[3] 3. Atiyah, M. F., R. Bott and A. Shapiro, Clifford Modules, Topology 3, (Supplement 1) (1964), 3–38. Google Scholar

[4] 4. Hoffman, P. N., r-Rings and Wreath Product Representations, Lecture Notes in Mathematics 746, Springer-Verlag, Berlin, 1979. Google Scholar

[5] 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] 6. Hoffman, P. N., A Projective Analogue of Schur's Tensor Power Construction, Comm. Algebra 21(1993), 2211—2249. Google Scholar

[7] 7. Hoffman, P. N., Generalizing Clifford Module Periodicity to Graded Representations, (1992), preprint. Google Scholar

[8] 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] 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] 10. Hoffman, P. N. and Humphreys, J. F., Projective Representations of the Symmetric Groups, Oxford University Press, Oxford, 1992. Google Scholar

[11] 11. Huang, J. Q., Ph.D. Thesis, University of Waterloo, 1993, 1–128. Google Scholar

[12] 12. James, G. D., The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics 682, Springer-Verlag, Berlin, 1978. Google Scholar

[13] 13. Jôzefiak, T., Characters of Projective Representations of Symmetric Groups, Exposition. Math. 7(1989), 193–247. Google Scholar

[14] 14. Karoubi, M., K-Theory : An Introduction, Springer-Verlag, Berlin, Heidleberg, New York, 1977. Google Scholar

[15] 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] 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] 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] 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] 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] 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] 21. Serre, J. P., Linear Representations of Finite Groups, Graduate Text in Math. 42, Springer-Verlag, New York, 1977. Google Scholar

[22] 22. Stembridge, J. R., Shifted Tableaux and the Projective Representations of Symmetric Groups, Adv. in Math. 74(1989), 87–134. Google Scholar