A note on the hit problem for the polynomial algebra of six variables
and the sixth algebraic transfer

Let $\mathcal A$ be the classical, singly-graded
Steenrod algebra over the prime order field $\mathbb
F_2$ and let $P^{\otimes h}: =
\mathbb F_2[t_1, \ldots, t_h]$
denote the polynomial algebra on $h$ generators, each of degree $1$,
viewed as a module over $\mathcal A.$ Write $GL_h$
for the usual general linear group of rank $h$ over
$\mathbb F_2.$ As well known, the (mod 2) cohomology
groups of the Steenrod algebra, ${\rm
Ext}_{\mathcal A}^{h,
h+*}(\mathbb F_2, \mathbb F_2)$ is
still largely mysterious for all homological degrees $h
\geq 6.$ The $h$-th algebraic transfer
$$Tr_h^{\mathcal A}: (\mathbb
F_2\otimes_{GL_h}{\rm
Ann}_{\overline{\mathcal
A}}[P^{\otimes
h}]^{*})_n\longrightarrow {\rm
Ext}_{\mathcal A}^{h,
h+n}(\mathbb F_2, \mathbb F_2),$$
defined by William Singer \cite{Singer}, is a helpful
tool to describe that Ext groups. Singer conjectured that this transfer
is a monomorphism, but it remains open for any $h\geq
5.$ There is currently no information on the conjecture for $h = 6$.
In this paper, we verify Singer’s conjecture for all homological degrees
$h\geq 1$ in the internal degrees $\leq
10 = 6(2^{0}-1) + 10\cdot 2^{0}$ and for $h =
6$ in the degree of the general form $n_s:=6(2^{s}-1) +
10\cdot 2^{s}.$ This result is important, since it
tells us that the non-zero elements $h_2^{2}g_1 =
h_4Ph_2\in {\rm
Ext}_{\mathcal A}^{6,
6+n_1}(\mathbb F_2, \mathbb F_2)$,
and $D_2\in {\rm
Ext}_{\mathcal A}^{6,
6+n_2}(\mathbb F_2, \mathbb F_2)$ are
not in the image of the sixth algebraic transfer. This Note is a
continuation of our previous one \cite{Phuc11}, which
will refer to as Part I. Our approach is based on explicitly solving the
hit problem for the Steenrod algebra in the case $h = 6$ and degree
$n_s.$ This extends a result in Mothebe, Kaelo and Ramatebele
\cite{MKR}. At the same time, we also use this obtained
result to establish the dimension result for the space of the unhit
elements, $\mathbb
F_2\otimes_{\mathcal
A}P^{\otimes 7}$ in a certain general degree.

Jan 2023