Proceedings of the Japan Academy, Series A, Mathematical Sciences

The squaring operation on ${\cal A}$-generators of the Dickson algebra

Nguyên H. V. Hưng and Võ T. N. Quỳnh

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Abstract

We study the squaring operation $Sq^0$ on the dual of the minimal ${\cal A}$-generators of the Dickson algebra. We show that this squaring operation is isomorphic on its image. We also give vanishing results for this operation in some cases. As a consequence, we prove that the Lannes-Zarati homomorphism vanishes (1) on every element in any finite $Sq^0$-family in $Ext_{\cal A}^*({\bf F}_2, {\bf F}_2)$ except possibly the family initial element, and (2) on almost all known elements in the Ext group. This verifies a part of the algebraic version of the classical conjecture on spherical classes.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 85, Number 6 (2009), 67-70.

Dates
First available in Project Euclid: 3 June 2009

Permanent link to this document
https://projecteuclid.org/euclid.pja/1244037799

Digital Object Identifier
doi:10.3792/pjaa.85.67

Mathematical Reviews number (MathSciNet)
MR2532421

Zentralblatt MATH identifier
1175.55008

Subjects
Primary: 55P47: Infinite loop spaces 55Q45: Stable homotopy of spheres 55S10: Steenrod algebra 55T15: Adams spectral sequences

Keywords
Modular representations invariant theory cohomology of the Steenrod algebra spherical classes Lannes-Zarati homomorphism

Citation

Hưng, Nguyên H. V.; Quỳnh, Võ T. N. The squaring operation on ${\cal A}$-generators of the Dickson algebra. Proc. Japan Acad. Ser. A Math. Sci. 85 (2009), no. 6, 67--70. doi:10.3792/pjaa.85.67. https://projecteuclid.org/euclid.pja/1244037799


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