The cohomology of the Steenrod algebra and representations of the general linear groups

Author:
Nguyên H. V. Hung

Journal:
Trans. Amer. Math. Soc. **357** (2005), 4065-4089

MSC (2000):
Primary 55P47, 55Q45, 55S10, 55T15

DOI:
https://doi.org/10.1090/S0002-9947-05-03889-4

Published electronically:
May 20, 2005

MathSciNet review:
2159700

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be the algebraic transfer that maps from the coinvariants of certain -representations to the cohomology of the Steenrod algebra. This transfer was defined by W. Singer as an algebraic version of the geometrical transfer . It has been shown that the algebraic transfer is highly nontrivial, more precisely, that is an isomorphism for and that is a homomorphism of algebras.

In this paper, we first recognize the phenomenon that if we start from any degree and apply repeatedly at most times, then we get into the region in which all the iterated squaring operations are isomorphisms on the coinvariants of the -representations. As a consequence, every finite -family in the coinvariants has at most nonzero elements. Two applications are exploited.

The first main theorem is that is not an isomorphism for . Furthermore, for every , there are infinitely many degrees in which is not an isomorphism. We also show that if detects a nonzero element in certain degrees of , then it is not a monomorphism and further, for each , is not a monomorphism in infinitely many degrees.

The second main theorem is that the elements of any -family in the cohomology of the Steenrod algebra, except at most its first elements, are either all detected or all not detected by , for every . Applications of this study to the cases and show that does not detect the three families , and , and that does not detect the family .

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Additional Information

**Nguyên H. V. Hung**

Affiliation:
Department of Mathematics, Vietnam National University, Hanoi 334 Nguyễn Trãi Street, Hanoi, Vietnam

Email:
nhvhung@vnu.edu.vn

DOI:
https://doi.org/10.1090/S0002-9947-05-03889-4

Keywords:
Adams spectral sequences,
Steenrod algebra,
modular representations,
invariant theory

Received by editor(s):
November 13, 2003

Published electronically:
May 20, 2005

Additional Notes:
This work was supported in part by the National Research Program, Grant No. 140 804

Dedicated:
Dedicated to Professor Nguyễn Hữu Anh on the occasion of his sixtieth birthday

Article copyright:
© Copyright 2005
by Nguyễn H. V. Hưng, Nguyễn H. V. Khuê and Nguyễn My Trang