Title of Paper: The hit problem for the modular invariants of
linear groups
Author: Nguy\^{e}n H. V. Hung and Tran Ngoc Nam
2000 Mathematics Subject Classification: Primary 55S10, Secondary
55Q45.
Address of authors: Department of Mathematics, Vietnam National
University, Hanoi, 334 Nguyen Trai Street, Hanoi, Vietnam
E-mail address: nhvhung@vnu.edu.vn
E-mail address: namtn@vnu.edu.vn
Abstract: Let the mod 2 Steenrod algebra, ${\cal A}$, and the
general linear group, $GL_k:=GL(k, F_2)$, act on
$P_{k}:=F_2[x_{1},...,x_{k}]$ with $\deg(x_{i})=1$ in the usual
manner. We prove that, for a family of some rather small subgroups
$G$ of $GL_k$, every element of positive degree in the invariant
algebra $P_{k}^G$ is hit by ${\cal A}$ in $P_{k}$. In other words,
$(P_{k}^G)^+ \subset {\cal A}^+\cdot P_{k}$, where $(P_{k}^G)^+$
and ${\cal A}^+$ denote respectively the submodules of $P_{k}^G$
and ${\cal A}$ consisting of all elements of positive degree. This
family contains most of the parabolic subgroups of $GL_k$. It
should be noted that the smaller the group G is the harder the
problem turns out to be. Remarkably, when $G$ is the smallest
group of the family, the invariant algebra $P_{k}^G$ is a
polynomial algebra in $k$ variables, whose degrees are $\leq 8$
and fixed while $k$ increases.
It has been shown by Hung [Trans AMS 349 (1997), 3893-3910] that,
for $G=GL_k$, the inclusion $(P_{k}^{GL_k})^+\subset {\cal
A}^+\cdot P_{k}$ is equivalent to a week algebraic version of the
long-standing conjecture stating that the only spherical classes
in $Q_0S^0$ are the elements of Hopf invariant one and those of
Kervaire invariant one.