Abstract
We prove that the mod 2 complete Steenrod algebra  is closely related to the Dickson algebra, the invariant algebra of GL(k, ℤ/2). More precisely,  is dual to \( D_{\infty }^{{\sqrt {{}} }} \), the generalized Dickson algebra on infinitely many generators, as a \( {\mathcal Z}\lbrack\frac{1}{2}\rbrack \)-graded algebra. We also show that the generalized operations in  are derived from the generalized Dickson invariants in a similar way as the operations in A are derived from the Dickson invariants (see Mùi [5], Madsen-Milgram [9], Lomonaco [7]).
The first-named author was partially supported by the DGICYT, PB 91-0467. The second-named author was supported by the DGU- through the CRM (Barcelona).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
D. Arnon, Generalized Dickson invariants, Ph. D. Thesis, MIT, 1994.
S. R. Bullett and I. G. Macdonald, On the Adem relations, Topology 21 (1982), 329–332.
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.
Huýnh Mùi, Modular invariant theory and the cohomology algebras of symmetric groups, J. Fac. Sci. Univ. Tokyo Sec. IA Math. 22 (1975), 319–369.
Huýnh Mùi, Dickson invariants and Milnor basis of the Steenrod algebra, Eger Internat. Colloq. Topology (1983), 345–355.
J. Lannes and S. Zarati, Sur les foncteurs dérivés de la déstabilisation, Math. Zeit. 194 (1987), 25–59.
L. Lomonaco, The iterated total squaring operation, Proc. Amer. Math. Soc. 115 (1992), 1149–1155.
I. Madsen, On the action of the Dyer-Lashof algebra in H *(G), Pacific J. Math. 60 (1975), 235–275.
I. Madsen and J. Milgram, The classifying spaces for surgery and cobordism of manifolds, Ann. of Math. Studies, No. 92, Princeton Univ. Press, 1979.
J. Milnor, The Steenrod algebra and its dual, Ann. of Math. 67 (1958), 150–171.
Nguyên N. Hai and Nguyên H. V. Hu’ng, Steenrod operations on mod 2 homology of the iterated loop space, Acta Math. Vietnam. 13 (1988), 113–126.
Nguyên H. V. Hu’ng, The action of the Steenrod squares on the modular invariants of linear groups, Proc. Amer. Math. Soc. 113 (1991), 1097–1104.
Nguyên H. V. Hu’ng, Spherical classes and the algebraic transfer, CRM Preprint.
Nguyên H. V. Hu’ng and F. P. Peterson, A-generators for the Dickson algebra, Trans. Amer. Math. Soc. (to appear).
F. P. Peterson, Private communication.
W. M. Singer, A new chain complex for the homology of the Steenrod algebra, Proc. Cambridge Philos. Soc. 90 (1981), 279–292.
W. M. Singer, Invariant theory and the lambda algebra, Trans. Amer. Math. Soc. 280 (1983), 673–693.
N. E. Steenrod and D. B. A. Epstein, Cohomology operations, Ann. of Math. Studies, No. 50, Princeton Univ. Press, 1962.
C. Wilkerson, Classifying spaces, Steenrod operations and algebraic closure, Topology 16 (1977), 227–237.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Birkhäuser Verlag
About this paper
Cite this paper
Llerena, I., Hu’ng, N.H.V. (1996). The complete Steenrod algebra and the generalized Dickson algebra. In: Broto, C., Casacuberta, C., Mislin, G. (eds) Algebraic Topology: New Trends in Localization and Periodicity. Progress in Mathematics, vol 136. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9018-2_21
Download citation
DOI: https://doi.org/10.1007/978-3-0348-9018-2_21
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0348-9869-0
Online ISBN: 978-3-0348-9018-2
eBook Packages: Springer Book Archive