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The action of the Steenrod squares on the modular invariants of linear groups


Author: Nguyên H. V. Hung
Journal: Proc. Amer. Math. Soc. 113 (1991), 1097-1104
MSC: Primary 55S05
DOI: https://doi.org/10.1090/S0002-9939-1991-1064904-6
MathSciNet review: 1064904
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Abstract: We compute the action of the Steenrod squares on the Dickson invariants of the group $ G{L_n} = GL(n,{\mathbf{Z}}/2)$ and the Mùi invariants of the subgroup $ {T_n}$ consisting of all upper triangular matrices with 1 on the main diagonal. Our method is very elementary. Roughly speaking, we read off the above action from the expansion of the Mùi invariants in terms of Dickson and Mùi invariants of fewer variables.


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  • [1] H. E. A. Campbell, Upper triangular invariants, Canad. Math. Bull. 28 (1985), 243-248. MR 782784 (86i:55023)
  • [2] L. E. Dickson, 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. MR 1500882
  • [3] Huynh Mùi, Modular invariant theory and cohomology algebras of symmetric groups, J. Fac. Sci. Univ. Tokyo Sec. IA 22 (1975), 319-369. MR 0422451 (54:10440)
  • [4] -, Dickson invariants and Milnor basis of the Steenrod algebra, Eger Internat. Colloq. Topology (1983), 345-355. MR 863917 (88a:55019)
  • [5] I. Madsen, On the action of the Dyer-Lashof algebra in $ {H_ * }(G)$, Pacific J. Math. 60 (1975), 235-275. MR 0388392 (52:9228)
  • [6] I. Madsen and R. J. Milgram, The classifying spaces for surgery and cobordism of manifolds, Ann. of Math. Studies, No. 92, Princeton Univ. Press, 1979. MR 548575 (81b:57014)
  • [7] B. M. Mann, The cohomology of the symmetric groups, Trans. Amer. Math. Soc. 242 (1978), 157-184. MR 0500961 (58:18447)
  • [8] B. M. Mann and R. J. Milgram, On the Chern classes of the regular representations of some finite groups, Proc. Edinburgh Math. Soc. 25 (1982), 259-268. MR 678549 (84c:20060)
  • [9] J. P. May, F. R. Cohen, and T. J. Lada, The homology of iterated loop spaces, Lecture Notes in Math., vol. 533, Springer-Verlag, Berlin and New York, 1976. MR 0436146 (55:9096)
  • [10] J. Milnor, The Steenrod algebra and its dual, Ann. of Math. 67 (1958), 150-171. MR 0099653 (20:6092)
  • [11] Nguyên N. Hai and Nguyên H. V. Hung, Steenrod operations on $ \bmod 2$ homology of the iterated loop space, Acta Math. Vietnam. 13 (1988), 113-126. MR 1023710 (90m:55019)
  • [12] W. M. Singer, On the construction of certain algebras over the Steenrod algebra, J. Pure Appl. Algebra 11 (1977), 53-59. MR 0467746 (57:7598)
  • [13] -, The transfer in homological algebra, Math. Z. 202 (1989), 493-523. MR 1022818 (90i:55035)
  • [14] L. Smith and R. Switzer, Realizability and non-realizability of Dickson algebras as cohomology rings, Proc. Amer. Math. Soc. 89 (1983), 303-313. MR 712642 (85e:55036)
  • [15] N. E. Steenrod and D. B. A. Epstein, Cohomology operations, Ann. of Math. Studies, No. 50, Princeton Univ. Press, 1962. MR 0145525 (26:3056)
  • [16] C. Wilkerson, A primer on the Dickson invariants, Contemporary Mathematics, Amer. Math. Soc., Providence, R.I., vol. 19, 1983, pp. 421-434. MR 711066 (85c:55017)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1064904-6
Keywords: Modular invariants, Steenrod operations
Article copyright: © Copyright 1991 American Mathematical Society

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