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.
Publié le : 2009-06-15
Classification:
Modular representations,
invariant theory,
cohomology of the Steenrod algebra,
spherical classes,
Lannes-Zarati homomorphism,
55P47,
55Q45,
55S10,
55T15
@article{1244037799,
author = {Hung, Nguyen H. V. and Quynh, Vo T. N.},
title = {The squaring operation on ${\cal A}$-generators of the Dickson algebra},
journal = {Proc. Japan Acad. Ser. A Math. Sci.},
volume = {85},
number = {2},
year = {2009},
pages = { 67-70},
language = {en},
url = {http://dml.mathdoc.fr/item/1244037799}
}
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., Tome 85 (2009) no. 2, pp. 67-70. http://gdmltest.u-ga.fr/item/1244037799/