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Truncated symmetric powers and modular representations of GLn

Published online by Cambridge University Press:  24 October 2008

Stephen Doty  and
Grant Walker
Stephen Doty
Affiliation:
Department of Mathematical Sciences, Loyola University of Chicago, 6525 N. Sheridan Road, Chicago, Illinois 60626, U.S.A.
Grant Walker
Affiliation:
Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL
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Abstract

Several results are obtained relating to the modular representation theory of the general linear group GLn in the defining characteristic p > 0. In Section 1, embeddings of certain simple modules in symmetric powers of the natural module, or in tensor products of truncated symmetric powers, are constructed. In Section 2, cases are found where simple quotientsof Schur modules H0(λ) can be constructed by extending theidea of truncation to these modules in a natural way. In Section 3, the characters of those simple modules which can be constructed as twisted tensor products of truncated symmetric powers are expressed in terms of Schur functions.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 119 , Issue 2 , February 1996 , pp. 231 - 242
Copyright
Copyright © Cambridge Philosophical Society 1996

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References

REFERENCES

[1]Carlisle, D. P., Eccles, P., Hilditch, S., Ray, N., Schwartz, L., Walker, G. and Wood, R.. Modular representations of GLn(p), splitting Σ(CP∞ × … × CP∞), and the β-family as framed hypersurfaces. Math. Zeit. 189 (1985), 239–261.CrossRefGoogle Scholar
[2]Carlisle, D. P. and Kuhn, N. J.. Subalgebras of the Steenrod algebra and the action of matrices on truncated polynomial algebras. J. Algebra 121 (1989), 370–387.CrossRefGoogle Scholar
[3]Carlisle, D. P. and Walker, G.. Poincaré series for the occurrence of certain modular representations of GLn(p) in the symmetric algebra. Proc. Royal Soc. Edinburgh 113A (1989), 27–41.CrossRefGoogle Scholar
[4]Donkin, S.. On Schur algebras and related algebras II. J. Algebra 111 (1987), 354–364.CrossRefGoogle Scholar
[5]Doty, S. R.. The submodule structure of certain Weyl modules for groups of type An. J. Algebra 95 (1985), 373–383.CrossRefGoogle Scholar
[6]Doty, S. R.. Submodules of symmetric powers of the natural module for GLn. Contemporary Math. 88 (1989), 185–191.CrossRefGoogle Scholar
[7]Doty, S. R. and Walker, G.. Modular symmetric functions and irreducible modular representations of general linear groups. J. Pure and Applied Algebra 82 (1992), 1–26.CrossRefGoogle Scholar
[8]Green, J. A.. Polynomial Representations of GLn. Lecture Notes in Mathematics 830 (Springer-Verlag, 1980).Google Scholar
[9]Harris, J. C. and Kuhn, N. J.. Stable decompositions of classifying spaces of finite abelian p–groups. Math. Proc. Cambridge Philos. Soc. 103 (1988), 427–449.CrossRefGoogle Scholar
[10]James, G. D.. The decomposition of tensors over fields of prime characteristic. Math. Zeit. 172 (1980), 161–178.CrossRefGoogle Scholar
[11]James, G. D. and Kerber, A.. The representation theory of the symmetric group. Encyclopaedia of Mathematics and its Applications vol. 16 (Addison-Wesley, 1981).Google Scholar
[12]Jantzen, J. C.. Representations of algebraic groups. (Academic Press, 1987).Google Scholar
[13]Kouwenhoven, F. W.. Schur and Weyl functors II. Communications in Algebra 18 (1990), 2885–2941.CrossRefGoogle Scholar
[14]Krasón, P. and Kuhn, N. J.. On embedding polynomial functors in symmetric powers. J. Algebra 163 (1994), 281–294.CrossRefGoogle Scholar
[15]Krop, L.. On the representations of the full matrix semigroup on homogeneous polynomials. I, J. Algebra 99 (1986), 370–421; II, J. Algebra 102 (1986), 284–300.CrossRefGoogle Scholar
[16]Macdonald, I. G.. Symmetric functions and Hall polynomials. (Oxford University Press, 1979).Google Scholar
[17]Martin, S.. Schur algebras and representation theory. Cambridge Tracts in Mathematics no. 112 (Cambridge University Press, 1994).CrossRefGoogle Scholar
[18]Mitchell, S. A.. Finite complexes with A(n)-free cohomology. Topology 24 (1985), 227–248.CrossRefGoogle Scholar
[19]Mitchell, S. A. and Priddy, S. B.. Stable splittings derived from the Steinberg module. Topology 22 (1983), 285–298.CrossRefGoogle Scholar
[20]Mullineux, G.. Bijections of p–regular partitions and p–modular irreducibles of the symmetric groups. J. London Math. Soc. 20 (1979), 60–66.CrossRefGoogle Scholar
[21]Sullivan, J. B.. Some representation theory for the modular general linear groups. J. Algebra 45 (1977), 516–535.CrossRefGoogle Scholar
[22]Walker, G.. Modular Schur functions. Trans. Amer. Math. Soc. 346 (1994), 569–604. Amer.Math. Soc. 346 (1994), 569–604.CrossRefGoogle Scholar
[23]Wong, W. J.. Irreducible representations of Chevalley groups. J. Algebra 20 (1972), 355–367.CrossRefGoogle Scholar