Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The hit problem for the Dickson algebra


Authors: Nguyen H. V. Hu'ng and Tran Ngoc Nam
Journal: Trans. Amer. Math. Soc. 353 (2001), 5029-5040
MSC (2000): Primary 55S10; Secondary 55P47, 55Q45, 55T15.
DOI: https://doi.org/10.1090/S0002-9947-01-02705-2
Published electronically: May 22, 2001
MathSciNet review: 1852092
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract:

Let the mod 2 Steenrod algebra, $\mathcal{A}$, and the general linear group, $GL(k,{\mathbb{F} }_2)$, act on $P_{k}:={\mathbb{F} }_2[x_{1},...,x_{k}]$ with $\vert x_{i}\vert=1$ in the usual manner. We prove the conjecture of the first-named author in Spherical classes and the algebraic transfer, (Trans. Amer. Math Soc. 349 (1997), 3893-3910) stating that every element of positive degree in the Dickson algebra $D_{k}:=(P_{k})^{GL(k, {\mathbb{F} }_2)}$ is $\mathcal{A}$-decomposable in $P_{k}$ for arbitrary $k>2$. 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 $Q_0S^0$ are the elements of Hopf invariant one and those of Kervaire invariant one.


References [Enhancements On Off] (What's this?)

  • 1. 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. CMP 95:18
  • 2. Nguyen 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. MR 92c:55018
  • 3. Nguyen H. V. Hu'ng, Spherical classes and the algebraic transfer, Trans. Amer. Math. Soc. 349 (1997), 3893-3910. MR 98e:55020
  • 4. Nguyen H. V. Hu'ng, The weak conjecture on spherical classes, Math. Zeit. 231 (1999), 727-743. MR 2000g:55019
  • 5. Nguyen H. V. Hu'ng, Spherical classes and the lambda algebra, Trans. Amer. Math. Soc. (to appear).
  • 6. Nguyen H. V. Hu'ng and F. P. Peterson, ${\mathcal{A}}$-generators for the Dickson algebra, Trans. Amer. Math. Soc. 347 (1995), 4687-4728. MR 96c:55022
  • 7. Nguyen H. V. Hu'ng and F. P. Peterson, Spherical classes and the Dickson algebra, Math. Proc. Camb. Phil. Soc. 124 (1998), 253-264. MR 99i:55021
  • 8. M. Kameko, Products of projective spaces as Steenrod modules, Thesis, Johns Hopkins University 1990.
  • 9. F. P. Peterson, Generators of $H^*(\mathbf{RP}^{\infty}\wedge\,\mathbf{RP}^{\infty})$ 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), 737-748. MR 91i:55020
  • 11. J. H. Silverman, Hit polynomials and the canonical antiautomorphism of the Steenrod algabra, Proc. Amer. Math. Soc. 123 (1995), 627-637. MR 95c:55023
  • 12. W. M. Singer, The transfer in homological algebra, Math. Zeit. 202 (1989), 493-523. MR 90i:55035
  • 13. N. E. Steenrod and D. B. A. Epstein, Cohomology operations, Ann. of Math. Studies, No. 50, Princeton Univ. Press, 1962. MR 26:3056
  • 14. R. M. W. Wood, Modular representations of $GL(n,\mathbb{F} _p)$ and homotopy theory, Lecture Notes in Math. 1172, Springer Verlag (1985), 188-203. MR 88a:55007

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 55S10, 55P47, 55Q45, 55T15.

Retrieve articles in all journals with MSC (2000): 55S10, 55P47, 55Q45, 55T15.


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: https://doi.org/10.1090/S0002-9947-01-02705-2
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

American Mathematical Society