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Hopf Algebras and Projective Representations of G ≀ Sn AND G ≀ An

Published online by Cambridge University Press:  20 November 2018

Peter N. Hoffman  and
John F. Humphreys
Peter N. Hoffman
Affiliation:
University of Waterloo, Waterloo, Ontario
John F. Humphreys
Affiliation:
Liverpool University, Liverpool, England
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In 1911, Schur published a rather formidable paper [9] in which he determined all the complex projective characters for the symmetric group (denoted Σn here, despite the title), and for the alternating group An (A pronounced “alpha”). As far as we know, the construction of the modules involved is still an unsolved problem. The results of Schur can be expressed in terms of certain induced representations whose characters form a basis for the group of virtual characters, plus formulae expressing the irreducible characters in terms of these induced characters. Here we give a new formulation of the above induced characters in the spirit of the well known “induction algebra” approach to the linear representations of Σn. We use some Hopf algebra techniques inspired by [5] to give new proofs of Schur's results, and to determine the extra structure which we define.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 38 , Issue 6 , 01 December 1986 , pp. 1380 - 1458
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Atiyah, A., Bott, R. and Shapiro, A., Clifford modules, Topology 3 (Supp. 1) (1964), 3–38.Google Scholar
2. Dold, A., Lectures on algebraic topology (Springer-Verlag, 1972).CrossRefGoogle Scholar
3. Hoffman, P., τ-rings and wreath product representations, SLN 746 (Springer, 1979).CrossRefGoogle Scholar
4. Humphreys, John, Conjugacy classes of coverings of monomial groups (to appear).Google Scholar
5. Liulevicius, A., Arrows, symmetries and representation rings, J. Pure App. Algebra 19 (1980), 259–73.Google Scholar
6. MacDonald, I. G., Symmetric functions and Hall polynomials (Oxford U. Press, 1979).Google Scholar
7. Morris, A. O., Projective representations of reflection groups II, Proc. London Math. Soc. (3) 40 (1980), 553–76.Google Scholar
8. Read, E. W., On the projective characters of the symmetric group, J. London Math. Soc. (2) 75 (1977), 456–64.Google Scholar
9. Schur, I., Uber die Darstellung der symmetrischen und der alterierenden Gruppe durch gebrochene lineare Substitutionen, Collected Works, Vol. I (Springer-Verlag, 1973), 346–441.Google Scholar