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The Steenrod algebra action on generators of rings of invariants of subgroups of $ {\rm GL}\sb n({\bf Z}/p{\bf Z})$


Author: Nondas E. Kechagias
Journal: Proc. Amer. Math. Soc. 118 (1993), 943-952
MSC: Primary 55S10
DOI: https://doi.org/10.1090/S0002-9939-1993-1152986-4
MathSciNet review: 1152986
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Abstract: We compute the action of the Steenrod algebra on rings of invariants of certain subgroups of $ {\operatorname{GL} _n}({\mathbf{Z}}/p{\mathbf{Z}})$, namely, the upper triangular and parabolic subgroups. As a consequence we get the action on the Dickson algebra for any prime $ p$. The methods we use are elementary.


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  • [1] J. F. Adams and C. Wilkerson, Finite $ H$-spaces and algebras over the Steenrod algebra, Ann. of Math. (2) 111 (1980), 95-143. MR 558398 (81h:55006)
  • [3] A. Adem, J. Maginnis, and R. J. Milgram, Symmetric invariants and cohomology of groups, Ann. of Math. (2) 287 (1990), 391-411. MR 1060683 (91i:55022)
  • [3] H. E. A. Campbell, Upper triangular invariants, Canad. Math. Bull 28 (1985), 243-248. MR 782784 (86i:55023)
  • [4] H. E. A. Campbell and P. S. Selick, Polynomial algebras over the Steenrod algebra, Comment. Math. Helv. 64 (1990), 171-180. MR 1057238 (91f:55006)
  • [5] B. Cooper, Cohomology of symmetric groups, mimeo, Augsberg College, 1975.
  • [6] 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
  • [7] L. Evens, The cohomology ring of a finite group, Trans. Amer. Math. Soc. 101 (1961), 224-239. MR 0137742 (25:1191)
  • [8] Hyunh Múi, Modular invariant theory and cohomology algebras of symmetric groups, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), 319-369. MR 0422451 (54:10440)
  • [9] -, Homology operations derived from modular coinvariants, Lecture Notes in Math., vol. 172, Springer-Verlag, New York, 1985, pp. 85-115. MR 825776 (87g:55038)
  • [10] N. E. Kechagias, Ph.D. thesis, Queen's University, Kingston, Ontario, 1990.
  • [11] -, Mod odd modular coinvariants, homology operations, and limit spaces, preprint.
  • [12] I. Madsen, On the action of the Dyer-Lashof algebra in $ {H_{\ast}}(F)$, Pacific J. Math. 60 (1975), 235-275. MR 0388392 (52:9228)
  • [13] I. Madsen and R. J. Milgram, The classifying spaces for surgery and cobordism of manifolds, Ann. of Math. (2) 92 (1979). MR 548575 (81b:57014)
  • [14] B. N. Mann, The cohomology of the symmetric groups, Trans. Amer. Math. Soc. 242 (1978), 157-184. MR 0500961 (58:18447)
  • [15] J. P. May, The homology of $ {E^\infty }$-spaces, Lecture Notes in Math., vol. 533, Springer-Verlag, New York, 1975.
  • [16] Nguyén H. V. Hung, The action of the Steenrod squares on the modular invariants of linear groups, Proc. Amer. Math. Soc. 113 (1991), 1097-1104. MR 1064904 (92c:55018)
  • [17] 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)
  • [18] 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)
  • [19] N. E. Steenrod and D. B. A. Epstein, Cohomology operations, Ann. of Math. Stud., no. 50, Princeton Univ. Press, Princeton, NJ, 1962. MR 0145525 (26:3056)
  • [20] C. Wilkerson, A primer on the Dickson invariants, Contemp. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1983, pp. 421-434. MR 711066 (85c:55017)
  • [21] R. M. W. Wood, Steenrod squares of polynomials, London Math. Soc. Lecture Note Ser., vol. 139, Cambridge Univ. Press, Cambridge and New York, 1989, pp. 173-177. MR 1055877 (91c:55030)

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DOI: https://doi.org/10.1090/S0002-9939-1993-1152986-4
Keywords: Modular invariants, Steenrod operations
Article copyright: © Copyright 1993 American Mathematical Society

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