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Twisted Dickson–Mui invariants and the Steinberg module multiplicity

Published online by Cambridge University Press:  18 March 2011

JINKUI WAN  and
WEIQIANG WANG
JINKUI WAN
Affiliation:
Department of Mathematics, Beijing Institute of Technology, Beijing, 100081, P.R. China. e-mail: wjk302@gmail.com
WEIQIANG WANG
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, VA 22904, U.S.A. e-mail: ww9c@virginia.edu
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Abstract

We determine the invariants, with arbitrary determinant twists, of the parabolic subgroups of the finite general linear group GLn(q) acting on the tensor product of the symmetric algebra S•(V) and the exterior algebra ∧•(V) of the natural GLn(q)-module V. In addition, we obtain the graded multiplicity of the Steinberg module of GLn(q) in S•(V) ⊗ ∧•(V), twisted by an arbitrary determinant power.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 151 , Issue 1 , July 2011 , pp. 43 - 57
Copyright
Copyright © Cambridge Philosophical Society 2011

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References

REFERENCES

[1]Crabb, M.Dickson–Mui invariants. Bull. London. Math. Soc. 37 (2005), 846–856.CrossRefGoogle Scholar
[2]Curtis, C.The Steinberg character of a finite group with a (B, N)-pair. J. Algebra 4 (1966), 433–441.CrossRefGoogle Scholar
[3]Dickson, L.A fundamental system of invariants of the general modular linear group with a solution of the form problem. Trans. Amer. Math. Soc. 12 (1911), 75–98.CrossRefGoogle Scholar
[4]Hewett, T.Modular invariant theory of parabolic subgroups of GL n(q) and the associated Steenrod modules. Duke Math. J. 82 (1996), 91–102; Erratum, Duke Math. J. 97 (1999), 217.Google Scholar
[5]Humphreys, J.Modular representations of finite groups of Lie type. LMS Lecture Note Series 326 (Cambridge University Press, 2006).Google Scholar
[6]Kuhn, N. and Mitchell, S.The multiplicity of the Steinberg representation of GL n(q) in the symmetric algebra. Proc. Amer. Math. Soc. 96 (1986), 1–6.Google Scholar
[7]Minh, P. and Tùng, V.Modular invariants of parabolic subgroups of general linear groups. J. Algebra 232 (2000), 197–208.CrossRefGoogle Scholar
[8]Mitchell, S.Finite complexes with A(n)-free cohomolgy. Topology 24 (1985), 227–248.Google Scholar
[9]Mitchell, S. and Priddy, S.Stable splittings derived from the Steinberg module. Topology 22 (1983), 285–298.CrossRefGoogle Scholar
[10]Mui, H.Modular invariant theory and chomomogy algebras of symmetric groups. J. Fac. Sci. Univ. Tokyo, Sect. 1A Math. 22 (1975), 319–369.Google Scholar
[11]Smith, L.Polynomial invariants of finite groups. Res. Notes in Math. 6 (AK Peters, Ltd., Wellesley, MA, 1995).Google Scholar
[12]Solomon, L. The Steinberg character of a finite group with BN-pair. Theory of Finite Groups (Harvard Symposium) (Benjamin, Elmsford, N.Y., 1969), 213–221.Google Scholar
[13]Wilkerson, C.A primer on the Dickson invariants. Contemp. Math. 19 (1983), 421–434.CrossRefGoogle Scholar
[14]Wan, J. and Wang, W. The GL n(q)-module structure of the symmetric algebra around the Steinberg module. arXiv:1012.0406 (2010).Google Scholar