𝒜-generators for the Dickson algebra

Hu’ng, Nguyễn H. V.; Peterson, Franklin P.

doi:10.1090/S0002-9947-1995-1316852-X

$\scr A$-generators for the Dickson algebra
HTML articles powered by AMS MathViewer

by Nguyễn H. V. Hu’ng and Franklin P. Peterson PDF

Trans. Amer. Math. Soc. 347 (1995), 4687-4728 Request permission

Abstract:

Let ${D_k}$ denote the Dickson algebra in $k$ variables over the field of two elements. We study the problem of determining a minimal set of generators for ${D_k}$ as a module over the Steenrod algebra $\mathcal {A}$. This is easy for $k = 1$ and $2$. In this paper we answer this question for $k = 3$ and $4$ and give techniques which may help solve the problem for general $k$.

References

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
Jean Lannes and Saïd Zarati, Invariants de Hopf d’ordre supérieur et suite spectrale d’Adams, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 15, 695–698 (French, with English summary). MR 705694
Jean Lannes and Saïd Zarati, Sur les foncteurs dérivés de la déstabilisation, Math. Z. 194 (1987), no. 1, 25–59 (French). MR 871217, DOI 10.1007/BF01168004
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, DOI 10.2140/pjm.1975.60.235
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
Nguyên H. V. Hung, The action of the Steenrod squares on the modular invariants of linear groups, Proc. Amer. Math. Soc. 113 (1991), no. 4, 1097–1104. MR 1064904, DOI 10.1090/S0002-9939-1991-1064904-6
William M. Singer, The transfer in homological algebra, Math. Z. 202 (1989), no. 4, 493–523. MR 1022818, DOI 10.1007/BF01221587
Robert J. Wellington, The unstable Adams spectral sequence for free iterated loop spaces, Mem. Amer. Math. Soc. 36 (1982), no. 258, viii+225. MR 646741, DOI 10.1090/memo/0258
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