The hit problem for the Dickson algebra
Authors:
Nguyen H. V. Hu'ng and Tran Ngoc Nam
Journal:
Trans. Amer. Math. Soc. 353 (2001), 50295040
MSC (2000):
Primary 55S10; Secondary 55P47, 55Q45, 55T15.
Published electronically:
May 22, 2001
MathSciNet review:
1852092
Fulltext PDF Free Access
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Abstract: Let the mod 2 Steenrod algebra, , and the general linear group, , act on with in the usual manner. We prove the conjecture of the firstnamed author in Spherical classes and the algebraic transfer, (Trans. Amer. Math Soc. 349 (1997), 38933910) stating that every element of positive degree in the Dickson algebra is decomposable in for arbitrary . This conjecture was shown to be equivalent to a weak algebraic version of the classical conjecture on spherical classes, which states that the only spherical classes in are the elements of Hopf invariant one and those of Kervaire invariant one.
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 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), 7598. CMP 95:18
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 Nguyen H. V. Hu'ng, The action of the Steenrod squares on the modular invariants of linear groups, Proc. Amer. Math. Soc. 113 (1991), 10971104. MR 92c:55018
 3.
 Nguyen H. V. Hu'ng, Spherical classes and the algebraic transfer, Trans. Amer. Math. Soc. 349 (1997), 38933910. MR 98e:55020
 4.
 Nguyen H. V. Hu'ng, The weak conjecture on spherical classes, Math. Zeit. 231 (1999), 727743. MR 2000g:55019
 5.
 Nguyen H. V. Hu'ng, Spherical classes and the lambda algebra, Trans. Amer. Math. Soc. (to appear).
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 Nguyen H. V. Hu'ng and F. P. Peterson, generators for the Dickson algebra, Trans. Amer. Math. Soc. 347 (1995), 46874728. MR 96c:55022
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 Nguyen H. V. Hu'ng and F. P. Peterson, Spherical classes and the Dickson algebra, Math. Proc. Camb. Phil. Soc. 124 (1998), 253264. MR 99i:55021
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 M. Kameko, Products of projective spaces as Steenrod modules, Thesis, Johns Hopkins University 1990.
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 F. P. Peterson, Generators of as a module over the Steenrod algebra, Abstracts Amer. Math. Soc., No 833, April 1987.
 10.
 S. Priddy, On characterizing summands in the classifying space of a group, I, Amer. Jour. Math. 112 (1990), 737748. MR 91i:55020
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 J. H. Silverman, Hit polynomials and the canonical antiautomorphism of the Steenrod algabra, Proc. Amer. Math. Soc. 123 (1995), 627637. MR 95c:55023
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 W. M. Singer, The transfer in homological algebra, Math. Zeit. 202 (1989), 493523. MR 90i:55035
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 N. E. Steenrod and D. B. A. Epstein, Cohomology operations, Ann. of Math. Studies, No. 50, Princeton Univ. Press, 1962. MR 26:3056
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 R. M. W. Wood, Modular representations of and homotopy theory, Lecture Notes in Math. 1172, Springer Verlag (1985), 188203. MR 88a:55007
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Additional Information
Nguyen H. V. Hu'ng
Affiliation:
Department of Mathematics, Vietnam National University, Hanoi, 334 Nguyên Trãi Street, Hanoi, Vietnam
Email:
nhvhung@hotmail.com
Tran Ngoc Nam
Affiliation:
Department of Mathematics, Vietnam National University, Hanoi, 334 Nguyên Trãi Street, Hanoi, Vietnam
Email:
trngnam@hotmail.com
DOI:
http://dx.doi.org/10.1090/S0002994701027052
PII:
S 00029947(01)027052
Keywords:
Steenrod algebra,
invariant theory,
Dickson algebra.
Received by editor(s):
September 29, 1999
Received by editor(s) in revised form:
February 22, 2000
Published electronically:
May 22, 2001
Additional Notes:
This work was supported in part by the National Research Project, No. 1.4.2
Dedicated:
Dedicated to Professor Franklin P. Peterson on the occasion of his 70th birthday
Article copyright:
© Copyright 2001
American Mathematical Society
