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

Hung, Nguyên H. V.

doi:10.1090/S0002-9939-1991-1064904-6

The action of the Steenrod squares on the modular invariants of linear groups
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by Nguyên H. V. Hung

Proc. Amer. Math. Soc. 113 (1991), 1097-1104

DOI: https://doi.org/10.1090/S0002-9939-1991-1064904-6

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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.

References

H. E. A. Campbell, Upper triangular invariants, Canad. Math. Bull. 28 (1985), no. 2, 243–248. MR 782784, DOI 10.4153/CMB-1985-030-2
Leonard Eugene 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), no. 1, 75–98. MR 1500882, DOI 10.1090/S0002-9947-1911-1500882-4
Huỳnh Mui, Modular invariant theory and cohomology algebras of symmetric groups, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), no. 3, 319–369. MR 422451
Huỳnh Mùi, Dickson invariants and Milnor basis of the Steenrod algebra, Topology, theory and applications (Eger, 1983) Colloq. Math. Soc. János Bolyai, vol. 41, North-Holland, Amsterdam, 1985, pp. 345–355. MR 863917
Ib Madsen, On the action of the Dyer-Lashof algebra in $H_{\ast }(G)$, Pacific J. Math. 60 (1975), no. 1, 235–275. MR 388392
Ib Madsen and R. James Milgram, The classifying spaces for surgery and cobordism of manifolds, Annals of Mathematics Studies, No. 92, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1979. MR 548575
Benjamin Michael Mann, The cohomology of the symmetric groups, Trans. Amer. Math. Soc. 242 (1978), 157–184. MR 500961, DOI 10.1090/S0002-9947-1978-0500961-9
Benjamin M. Mann and R. James Milgram, On the Chern classes of the regular representations of some finite groups, Proc. Edinburgh Math. Soc. (2) 25 (1982), no. 3, 259–268. MR 678549, DOI 10.1017/S0013091500016746
Frederick R. Cohen, Thomas J. Lada, and J. Peter May, The homology of iterated loop spaces, Lecture Notes in Mathematics, Vol. 533, Springer-Verlag, Berlin-New York, 1976. MR 0436146
John Milnor, The Steenrod algebra and its dual, Ann. of Math. (2) 67 (1958), 150–171. MR 99653, DOI 10.2307/1969932
Nguyen Nam Hai and Nguyễn Hữu Việt Hưng, Steenrod operations on mod $2$ homology of the iterated loop space, Acta Math. Vietnam. 13 (1988), no. 2, 113–126 (1989). MR 1023710
William M. Singer, The construction of certain algebras over the Steenrod algebra, J. Pure Appl. Algebra 11 (1977/78), no. 1-3, 53–59. MR 467746, DOI 10.1016/0022-4049(77)90039-1
William M. Singer, The transfer in homological algebra, Math. Z. 202 (1989), no. 4, 493–523. MR 1022818, DOI 10.1007/BF01221587
Larry Smith and R. M. Switzer, Realizability and nonrealizability of Dickson algebras as cohomology rings, Proc. Amer. Math. Soc. 89 (1983), no. 2, 303–313. MR 712642, DOI 10.1090/S0002-9939-1983-0712642-4
N. E. Steenrod, Cohomology operations, Annals of Mathematics Studies, No. 50, Princeton University Press, Princeton, N.J., 1962. Lectures by N. E. Steenrod written and revised by D. B. A. Epstein. MR 0145525
Clarence Wilkerson, A primer on the Dickson invariants, Proceedings of the Northwestern Homotopy Theory Conference (Evanston, Ill., 1982) Contemp. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1983, pp. 421–434. MR 711066, DOI 10.1090/conm/019/711066