# A note on the modular representation on the $\mathbb{Z}/2$-homology groups
of the fourth power of real projective space and its application

D-ặng Võ Phúc  Faculty of Education Studies, University of Khanh Hoa,
01 Nguyen Chanh, Nha Trang, Khanh Hoa, Viet Nam<EMAIL_ADDRESS>
###### Abstract

We write $BV_{h}$ for the classifying space of the elementary Abelian 2-group
$V_{h}$ of rank $h,$ which is homotopy equivalent to the cartesian product of
$h$ copies of $\mathbb{R}P^{\infty}.$ Its cohomology with
$\mathbb{Z}/2$-coefficients can be identified with the graded unstable algebra
$P^{\otimes h}=\mathbb{Z}/2[t_{1},\ldots,t_{h}]=\\{P^{\otimes h}_{n}\\}_{n\geq
0}$ over the Steenrod ring $\mathcal{A}$, where grading is by the degree of
the homogeneous terms $P^{\otimes h}_{n}$ of degree $n$ in $h$ generators with
the degree of each $t_{i}$ being one. Let $GL_{h}$ be the usual general linear
group of rank $h$ over $\mathbb{Z}/2.$ The algebra $P^{\otimes h}$ admits a
left action of $\mathcal{A}$ as well as a right action of $GL_{h}.$ A central
problem of homotopy theory is to determine the structure of the space of
$GL_{h}$-coinvariants, $\mathbb{Z}/2\otimes_{GL_{h}}{\rm
Ann}_{\overline{\mathcal{A}}}H_{n}(BV_{h};\mathbb{Z}/2),$ where ${\rm
Ann}_{\overline{\mathcal{A}}}H_{n}(BV_{h};\mathbb{Z}/2)={\rm
Ann}_{\overline{\mathcal{A}}}[P^{\otimes h}_{n}]^{*}$ denotes the space of
primitive homology classes, considered as a representation of $GL_{h}$ for all
$n.$ Solving this problem is very difficult and still open for $h\geq 4.$

In this Note, our intent is of studying the dimension of
$\mathbb{Z}/2\otimes_{GL_{h}}{\rm Ann}_{\overline{\mathcal{A}}}[P^{\otimes
h}_{n}]^{*}$ for the case $h=4$ and the ”generic” degrees $n$ of the form
$n_{k,r,s}=k(2^{s}-1)+r.2^{s},$ where $k,\,r,\,s$ are positive integers.
Applying the results, we investigate the behavior of the Singer cohomological
”transfer” of rank $4.$ Singer’s transfer is a homomorphism from a certain
subquotient of the divided power algebra
$\Gamma(a_{1}^{(1)},\ldots,a_{h}^{(1)})$ to mod-2 cohomology groups ${\rm
Ext}_{\mathcal{A}}^{h,h+*}(\mathbb{Z}/2,\mathbb{Z}/2)$ of the algebra
$\mathcal{A}.$ Singer’s algebraic transfer is one of the relatively efficient
tools in determining the cohomology of the Steenrod algebra. Additionally, in
higher ranks, by using the results on $\mathcal{A}$-generators for $P^{\otimes
5}$ and $P^{\otimes 6},$ we show in Appendix that the transfer of rank 5 is an
isomorphism in som certain degrees of the form $n_{k,r,s}$, and that the
transfer of rank 6 does not detect the non-zero elements
$h_{2}^{2}g_{1}=h_{4}Ph_{2}\in{\rm
Ext}_{\mathcal{A}}^{6,6+n_{6,10,1}}(\mathbb{Z}/2,\mathbb{Z}/2)$, and
$D_{2}\in{\rm Ext}_{\mathcal{A}}^{6,6+n_{6,10,2}}(\mathbb{Z}/2,\mathbb{Z}/2).$
Besides, we also probe the behavior of the Singer transfer of ranks 7 and 8 in
internal degrees $\leq 15.$

###### keywords:

Adams spectral sequences, Primary cohomology operations, Steenrod algebra,
lambda algebra, Peterson hit problem, Actions of groups on commutative rings,
Algebraic transfer

###### MSC:

[2010] 55Q45, 13A50, 55S10, 55S05, 55T15, 55R12

###### Contents

  1. 1 Introduction
  2. 2 Main results
    1. 2.1 The degree $n_{s}:=5.2^{s}-2$
    2. 2.2 The degree $n^{\prime}_{s}:=17.2^{s}-2$
  3. 3 Proofs of main results
    1. 3.1 The degree $n_{s}=5.2^{s}-2$
    2. 3.2 The degree $n^{\prime}_{s}=17.2^{s}-2$
  4. 4 Appendix
    1. 4.1 The behavior of $Tr_{5}^{\mathcal{A}}$ in some internal degrees
    2. 4.2 The behavior of $Tr_{6}^{\mathcal{A}}$ in some internal degrees
    3. 4.3 The behavior of $Tr_{7}^{\mathcal{A}},$ and $Tr_{8}^{\mathcal{A}}$ in internal degrees $\leq 15$

##  1 Introduction

The hit problem and its dual. Let $V_{h}$ be an elementary Abelian 2-group of
rank $h$, which it will usually be more useful to consider as an
$h$-dimensional vector space over the prime field $\mathbb{Z}/2$ of
characteristic $2$. Denote by $BV_{h}$ the classifying space of $V_{h},$ which
is homotopy equivalent to the cartesian product of $h$ copies of the infinite
dimensional projective space, $\mathbb{R}P^{\infty}.$ Writing $\mathcal{A}$
for the Steenrod algebra over $\mathbb{Z}/2,$ which is defined to be the
algebra of stable cohomology operations for singular cohomology with
coefficients in $\mathbb{Z}/2.$ The algebra $\mathcal{A},$ which acts on the
cohomology ring of a space, is widely studied by mathematicians whose
interests range from Algebraic topology and homotopy theory, to manifold
theory, combinatorics, and more. Let $P^{\otimes
h}=\mathbb{Z}/2[t_{1},\ldots,t_{h}]=\\{P^{\otimes h}_{n}\\}_{n\geq 0}$ be the
graded unstable algebra over $\mathcal{A},$ where each $t_{i}$ has grading
one, and $P^{\otimes h}_{n}=\langle\\{f\in P^{\otimes h}|\,\mbox{$f$ is a
homogeneous polynomial of degree $n$}\\}\rangle.$ It has been shown (see [2])
that the ring of endomorphisms of the algebra $P^{\otimes h}$ over
$\mathcal{A}$ is isomorphic to $\mathbb{Z}/2[M_{h}(\mathbb{Z}/2)],$ the ring
of the semi-group ring of $h\times h$-matrices over $\mathbb{Z}/2$. So, we can
be identified $P^{\otimes h}$ with the $\mathbb{Z}/2$-cohomology of $BV_{h}.$
As known, the algebra $\mathcal{A}$ is generated by the Steenrod squares
$Sq^{i}$ ($i\geq 0$) and subject to Adem’s relations:
$Sq^{i}Sq^{j}=\sum_{0\leq
t\leq[i/2]}\binom{j-t-1}{i-2t}Sq^{i+j-t}Sq^{t},\,\,0<i<2j,$ where the binomial
coefficients are to be interpreted mod 2. The action of $\mathcal{A}$ over the
polynomial algebra $P^{\otimes h}$ is determined by instability axioms. By
Cartan’s formula, it suffices to determine $Sq^{i}(t_{j})$ and the instability
axioms give $Sq^{1}(t_{j})=t_{j}^{2}$ while $Sq^{i}(t_{j})=0$ if $i>1.$ One of
the unsolved problems in Algebraic topology is to find minimal sets of
$\mathcal{A}$-generators for $P^{\otimes h}.$ It is the same as the problem of
investigating a basis for the space of the ”non-hit” elements
$\\{QP_{n}^{\otimes h}:=P_{n}^{\otimes h}/\overline{\mathcal{A}}P_{n}^{\otimes
h}=(\mathbb{Z}/2\otimes_{\mathcal{A}}P^{\otimes h})_{n}\\}_{n\geq 0},$ where
$\overline{\mathcal{A}}P_{n}^{\otimes h}:=P_{n}^{\otimes
h}\cap\overline{\mathcal{A}}P^{\otimes h}$ and $\overline{\mathcal{A}}$
denotes the set of positive degree elements in $\mathcal{A}.$ This is called
”hit problem” in literature [31] and is known for all integers $h$ less than
or equal to $4$ (see [31], [18], [43, 44]). Though much work has been done
(see, for instance [32, 33, 34, 35, 36, 37, 39, 40], [27], [26], [47], [48,
49], [51]), solving the hit problem in the general case seems to be out of
reach with the present techniques, even when $h$ equal to $5$ and in some
certain generic degrees. Remarkably, by the works [18], [47], and [52], we
need only to study this problem in degrees $n$ of the form
$k(2^{s}-1)+r.2^{s},$ whenever $k,s,r$ are positive integers satisfying
$\mu(r)<k<h.$ Here $\mu(r)$ denotes the smallest integer $m$ such that $r$ can
be represented as $r=\sum_{1\leq i\leq m}(2^{u_{i}}-1),$ where $u_{i}>0.$
Several other aspects of the hit problem has been discovered in the recent
works of Ault-Singer [3], Inoue [13], Janfada-Wood [14], Pengelley-William
[30]. The dual of the hit problem is to determine a subring of elements of the
Pontrjagin ring $H_{*}(BV_{h};\mathbb{Z}/2)=[P^{\otimes h}]^{*},$ which is
mapped to zero by all Steenrod squares of positive degrees, frequently denoted
by ${\rm Ann}_{\overline{\mathcal{A}}}[P^{\otimes h}]^{*}.$

The Singer cohomological transfer. We knew that the $\mathbb{Z}/2$-cohomology
of the Steenrod algebra is the $E_{2}$-term for the (2-local) Adams spectral
sequence, whose abutment is the 2-component of the stable homotopy groups of
spheres. More specifically, the $E_{2}$-page of this spectral sequence may be
identified as ${\rm Ext}_{\mathcal{A}}^{*,*}(\mathbb{Z}/2,\mathbb{Z}/2).$ This
is what is meant by the aphorism ”the cohomology of the Steenrod algebra is an
approximation to the stable homotopy groups of spheres”. So, studying the
structure of the Adams $E_{2}$-terms becomes very important in homotopy
theory. The study of the hit problem and its dual is closely related to
describe these $E_{2}$-terms via a cohomological ”transfer” of William Singer
[41]. This transfer is constructed as follows. Consider the polynomial ring of
one variable, $P^{\otimes 1}=\mathbb{Z}/2[t_{1}].$ The canonical
$\mathcal{A}$-action on $P^{\otimes 1}$ is extended to an $\mathcal{A}$-action
on $\mathbb{Z}/2[t_{1},t_{1}^{-1}],$ the ring of finite Laurent series with
require that $Sq^{n}(t_{1}^{-1})=t_{1}^{n-1}$ for all $n\geq 1.$ Then,
$\mathcal{P}=\langle\\{t_{1}^{a}|\ a\geq-1\\}\rangle$ is
$\mathcal{A}$-submodule of $\mathbb{Z}/2[t_{1},t_{1}^{-1}].$ One has a short-
exact sequence:

(1) $0\to P^{\otimes
1}\xrightarrow{q}\mathcal{P}\xrightarrow{\pi}\Sigma^{-1}\mathbb{Z}/2,$

where $q$ is the inclusion and $\pi$ is given by $\pi(t_{1}^{a})=0$ if
$a\neq-1$ and $\pi(t_{1}^{-1})=1.$ Denote by $e_{1}$ the element in ${\rm
Ext}_{\mathcal{A}}^{1}(\Sigma^{-1}\mathbb{Z}/2,P^{\otimes 1}),$ which is
represented by the cocyle associated to (1). For each $i\geq 1,$ the short-
exact sequence

(2) $0\to P^{\otimes(i+1)}\cong P^{\otimes i}\otimes_{\mathbb{Z}/2}P^{\otimes
1}\xrightarrow{1\otimes_{\mathbb{Z}/2}q}P^{\otimes
i}\otimes_{\mathbb{Z}/2}\mathcal{P}\xrightarrow{1\otimes_{\mathbb{Z}/2}\pi}\Sigma^{-1}P^{\otimes
i},$

determines a class $(e_{1}\times P^{\otimes i})\in{\rm
Ext}_{\mathcal{A}}^{1}(\Sigma^{-1}P^{\otimes i},P^{\otimes(i+1)}).$ Then,
using the cross product and Yoneda product, we have the element

$e_{h}=(e_{1}\times P^{\otimes(h-1)})\circ(e_{1}\times
P^{\otimes(h-2)})\circ\cdots\circ(e_{1}\times P^{\otimes 1})\circ e_{1}\in{\rm
Ext}_{\mathcal{A}}^{h}(\Sigma^{-h}\mathbb{Z}/2,P^{\otimes h}).$

Let $\Delta(e_{1}\times P^{\otimes i}):{\rm
Tor}_{h-i}^{\mathcal{A}}(\mathbb{Z}/2,\Sigma^{-h}P^{\otimes i})\to{\rm
Tor}_{h-i-1}^{\mathcal{A}}(\mathbb{Z}/2,P^{\otimes(i+1)})$ be the connecting
homomorphism associated to (2). Then, we have a composition of the connecting
homomorphisms

$\overline{\varphi}^{\mathcal{A}}_{h}=\Delta(e_{1}\times
P^{\otimes(h-1)})\circ\Delta(e_{1}\times
P^{\otimes(h-2)})\circ\cdots\circ\Delta(e_{1}\times P^{\otimes
1})\circ\Delta(e_{1})$

from ${\rm Tor}_{h}^{\mathcal{A}}(\mathbb{Z}/2,\Sigma^{-h}\mathbb{Z}/2)$ to
${\rm Tor}_{0}^{\mathcal{A}}(\mathbb{Z}/2,P^{\otimes h})=QP^{\otimes
h}=\mathbb{Z}/2\otimes_{\mathcal{A}}P^{\otimes h}$, determined by
$\overline{\varphi}_{h}(z)=e_{h}\cap z$ for any $z\in{\rm
Tor}_{h}^{\mathcal{A}}(\mathbb{Z}/2,\Sigma^{-h}\mathbb{Z}/2).$ Here $\cap$
denotes the cap product in homology with $\mathbb{Z}/2$-coefficients. The
image of $\overline{\varphi}_{h}$ is a submodule of the invariant space
$[QP^{\otimes h}]^{GL_{h}}.$ Hence, $\overline{\varphi}^{\mathcal{A}}_{h}$
induces homomorphism $\varphi_{h}^{\mathcal{A}}:{\rm
Tor}_{h}^{\mathcal{A}}(\mathbb{Z}/2,\Sigma^{-h}\mathbb{Z}/2)\to[QP^{\otimes
h}]^{GL_{h}}.$ Because the supension $\Sigma^{-h}$ induces an isomorphism
${\rm Tor}_{h,n}^{\mathcal{A}}(\mathbb{Z}/2,\Sigma^{-h}\mathbb{Z}/2)\cong{\rm
Tor}_{h,h+n}^{\mathcal{A}}(\mathbb{Z}/2,\mathbb{Z}/2),$ we have the
homomorphism

$\varphi_{h}^{\mathcal{A}}:{\rm
Tor}_{h,h+n}^{\mathcal{A}}(\mathbb{Z}/2,\mathbb{Z}/2)\to[QP^{\otimes
h}_{n}]^{GL_{h}}.$

Then its dual $Tr_{h}^{\mathcal{A}}:\mathbb{Z}/2\otimes_{GL_{h}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n}^{\otimes h}]^{*}\to{\rm
Ext}_{\mathcal{A}}^{h,h+n}(\mathbb{Z}/2,\mathbb{Z}/2)$ is called the
cohomological transfer (or algebraic transfer) of rank $h$. This transfer can
be seen as an algebraic formulation of a stable transfer $B(V_{h})_{+}^{S}\to
S^{0}.$ It has been shown that $Tr_{h}^{\mathcal{A}}$ is an isomorphism for
$h$ less than or equal to $3$ (see the work of Singer [41] and of Boardman
[4]). In addition, the ”total” algebraic transfer
$\\{Tr_{h}^{\mathcal{A}}\\}_{h\geq 0}$ is an algebraic homomorphism (see also
[41]). These events show that $Tr_{h}^{\mathcal{A}}$ is highly nontrivial and
should be an useful tool to study the mysterious Ext groups. Specifically,
Singer [41] states the following yet left-open: $Tr_{h}^{\mathcal{A}}$ is a
monomorphism, for all positive integers $h$. One should note that by the works
of Minami [23, 24], hit problems and this transfer also make a useful
contribution for surveying the Kervaire invariant one problem. A review of
known results on this problem is necessary for the interested readers. The
Kervaire invariant was first introduced by Browder’s work [6], where he shows
that the classes $h_{j}^{2}\in{\rm
Ext}_{\mathcal{A}}^{2,2^{j+1}}(\mathbb{Z}/2,\mathbb{Z}/2)$ are the permanent
cycles in the classical Adams spectral sequence at the prime 2 [1], if and
only if smooth framed manifolds of Kervaire invariant one exist only in
dimensions $2^{j+1}-2.$ The hypothetical element in stable $2^{j+1}-2$-stem,
$\pi^{S}_{2^{j+1}-2},$ represented by such a framed manifold is denoted by
$\theta_{j}.$ These $\theta_{j}$ are known to be exist for $j\leq 5$ (see, for
instance Lin, and Mahowald [19]). Many open issues in Algebraic and
Differential topology depend on knowing whether or not the Kervaire invariant
one elements $\theta_{j}$ for $j\geq 6.$ In 2016, Hill, Hopkins, and Ravenel
[17] indicated that these elements do not exist for $j\geq 7$. The case $j=6$
is still no answer. Thus, smooth framed manifolds of Kervaire invariant one
therefore exist only in dimensions $2,6,14,30,62,$ and possibly $126.$

The lambda algebra. Beyond Singer’s algebraic transfer, the mod 2 lambda
algebra $\Lambda$ of six authors [5] is also an important tool for computing
the cohomology of $\mathcal{A}.$ This algebra considered as the $E_{1}$-term
of the Adams spectral sequence converging to the 2-component of the stable
homotopy groups of spheres. (Note that a spectral sequence arises from a
filtration of the dual chain complex and it provides an alternative way to
determine the cohomology of the dual chain complex.) Let us briefly review the
related concepts of this algebra. Firstly, we know that $\Lambda$ is an
associative differential bigraded algebra with generators
$\lambda_{n}\in\Lambda^{1,n}$ ($n\geq 0$) and the Adem relations

(3) $\lambda_{i}\lambda_{2i+n+1}=\sum_{j\geq
0}\binom{n-j-1}{j}\lambda_{i+n-j}\lambda_{2i+1+j}\ \ \ (i\geq 0,\ n\geq 0)$

with differential

(4) $\delta(\lambda_{n-1})=\sum_{j\geq
1}\binom{n-j-1}{j}\lambda_{n-j-1}\lambda_{j-1}\ \ \ (n\geq 1),$

where the binomial coefficients $\binom{n-j-1}{j}$ modulo $2.$ Now, for non-
negative integers $j_{1},\ldots,j_{h},$ the element
$\lambda_{j_{1}}\ldots\lambda_{j_{h}}$ in $\Lambda$ is said to be a monomial
of length $h$. Then, by the relations (3), the $\mathbb{Z}/2$-vector subspace
$\Lambda^{h,n}=\langle\\{\lambda_{j_{1}}\ldots\lambda_{j_{h}}|\ j_{m}\geq 0,\
1\leq m\leq h,\ \sum_{1\leq m\leq h}j_{m}=n\\}\rangle$ of $\Lambda$ has an
additive basis consisting of all admissible monomials of length $h$ (i.e.,
those of the form $\lambda_{j_{1}}\ldots\lambda_{j_{h}}$ where $j_{i}\leq
2j_{i+1}$ for all $0<i<h$.) Moreover, it is known that there is an
endomorphism $Sq^{0}$ of $\Lambda,$ determined by
$Sq^{0}(\lambda_{j_{1}}\ldots\lambda_{j_{h}})=\lambda_{2j_{1}+1}\ldots\lambda_{2j_{h}+1}.$
Further, this $Sq^{0}$ respects the relations in (3) and commutes with the
differential $\delta$ in (4). So, it induces the first Steenrod operation in
Ext groups (see Palmieri [29] for more details). It is known that in standard
notation, $H_{*}(V_{h};\mathbb{Z}/2)=H_{*}(BV_{h};\mathbb{Z}/2)=[P^{\otimes
h}]^{*}=\Gamma(x_{1}^{(1)},\ldots,x_{h}^{(1)}),$ the divided polynomial
algebra on $V_{h}$ over $\mathbb{Z}/2,$ generated by
$x_{1}^{(1)},\ldots,x_{h}^{(1)},$ where $x_{i}^{(1)}$ is linear dual to
$t_{i}.$ The (right) action of the Steenrod algebra on $[P^{\otimes h}]^{*}$
is determined by the usual Cartan formula and for $k>0,$

$(x^{(n)}_{i})Sq^{k}=\left\\{\begin{array}[]{ll}\binom{n-k}{k}x^{(n-k)}&\mbox{if
$2k<n$},\\\ 0&\mbox{otherwise}.\end{array}\right.$

This shows that $[P^{\otimes h}]^{*}$ has a natural right $\mathcal{A}$-module
structure. An interesting $\mathbb{Z}/2$-linear map $\psi_{h}$ from
$[P^{\otimes h}_{n}]^{*}$ to $\Lambda^{h,n},$ established by Chơn and Hà [10],
is determined by $\psi_{h}(x_{1}^{(j_{1})}\ldots
x_{h}^{(j_{h})})=\lambda_{j_{h}}$ if $h=1,$ and
$\psi_{h}(x_{1}^{(j_{1})}\ldots x_{h}^{(j_{h})})=\sum_{k\geq
j_{h}}\psi_{h-1}(((x_{1}^{(j_{1})}\ldots
x_{h-1}^{(j_{h-1})}))Sq^{k-j_{h}})\lambda_{k}$ otherwise. The authors
indicated that this homomorphism can be considered as a representation in the
algebra $\Lambda$ of the algebraic transfer, and that if $\zeta\in{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n}^{\otimes h}]^{*},$ then the image of
$\zeta$ under $\psi_{h}$ is a cycle in $\Lambda^{h,n}$ and is a representative
of the image of the class $[\zeta]$ under $Tr_{h}^{\mathcal{A}}.$

The structure of the coinvariant space $\mathbb{Z}/2\otimes_{GL_{h}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n}^{\otimes h}]^{*}$ and the behavior of the
Singer transfer have been proved surprisingly difficult by many researchers;
for instance the interested reader can see the works by Boardman [4], Chơn and
Hà [9, 10], Crabb-Hubbuck [11], Crossley [12], Hà [15], Hưng [16], Minami
[23], Nam [28], the present writer [33, 34, 36, 37, 38, 39, 40], Sum [46, 47],
etc. These problems are still unknown for $h\geq 4.$ In the rank 4 case, by
the work of Sum [43], it is sufficient to study them in the ”generic degrees”
$n$ of the following forms:

  1. (i)

$2^{s+1}-i$ for $1\leq i\leq 3,$

  2. (ii)

$2^{s+m+1}+2^{s+1}-3,$

  3. (iii)

$2^{s+m}+2^{s}-2,$

  4. (iv)

$2^{s+m+u}+2^{s+m}+2^{s}-3,$

for any positive integers $s,\,m,\,u$. It is easy to see that all these
degrees $n$ can be written as $n=k(2^{s}-1)+r.2^{s},$ where $k,\,r,\,s$ are
positive integers and $0<\mu(r)<k<4.$ The information on the behavior of
$Tr_{4}^{\mathcal{A}}$ in degrees of the form (i) is known by Sum [46]. The
remaining cases have been partially studied by the present author [38, 40].

In this work, we will continue to probe the problems above for the case of the
generic degree $n=2^{s+m}+2^{s}-2$ with $m\in\\{2,4\\}$ and $s$ an arbitrary
positive integer. Thence, we show that the Singer transfer is an isomorphism
in bidegree $(4,4+n).$ Moreover, in Appendix, basing the works by Bruner [7],
Mothebe, and Uys [25], Moetele, and Mothebe [27], Mothebe, Kaelo, and
Ramatebele [26], Lin [20], Sum [43, 44, 45, 47], Walker, and Wood [51], Tin
[48, 49], and our previous works [34, 35, 36], we would like to survey the
behavior of the transfer of ranks $>4$ in some internal degrees. Most notably,
in the rank 6 case, we shall show that the transfer $Tr_{6}^{\mathcal{A}}$
does not detect the non-zero elements:

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus 2.0pt&h_{1}Ph_{1}\in{\rm
Ext}_{\mathcal{A}}^{6,6+10}(\mathbb{Z}/2,\mathbb{Z}/2),\ \ \ \ \ \ \ \ \ \ \
h_{0}Ph_{2}\in{\rm Ext}_{\mathcal{A}}^{6,6+11}(\mathbb{Z}/2,\mathbb{Z}/2),\\\
&h_{2}^{2}g_{1}=h_{4}Ph_{2}\in{\rm
Ext}_{\mathcal{A}}^{6,6+26}(\mathbb{Z}/2,\mathbb{Z}/2),\ D_{2}\in{\rm
Ext}_{\mathcal{A}}^{6,6+58}(\mathbb{Z}/2,\mathbb{Z}/2),\end{array}$

and that in the rank 7 case, the transfer $Tr_{7}^{\mathcal{A}}$ does not
detect the non-zero element $h_{0}^{2}Ph_{2}=h_{1}^{2}Ph_{1}\in{\rm
Ext}_{\mathcal{A}}^{7,7+11}(\mathbb{Z}/2,\mathbb{Z}/2).$ The results for the
elements $h_{1}Ph_{1},\,h_{0}Ph_{2},$ and $h_{0}^{2}Ph_{2}=h_{1}^{2}Ph_{1}$
were also proved by Chơn, and Hà [9, 10] using other techniques. In addition,
in ranks $h=6,\,7,\,8,$ we refute the results by Moetele, and Mothebe [27] on
the dimensions of $QP_{13}^{\otimes h}$, from which we also indicate that the
Singer transfer is an isomorphism in bidegree $(h,h+13).$ Our method is based
on techniques of the hit problem of four variables in [43, 44] and a
representation in the lambda algebra of $Tr_{h}^{\mathcal{A}}$. The techniques
can be applied to study Singer’s transfer of a higher rank in some certain
generic degrees. Our approach is different from the ones of Singer [41], Hưng
[16], Chơn, and Hà [9, 10], Nam [28] in studying the image of the
cohomological transfer.

##  2 Main results

As mentioned above, and more precisely, we shall explicitly determine the
structure of the coinvariants $\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n}^{\otimes 4}]^{*}$ at the generic degrees
$n=5.2^{s}-2$ and $17.2^{s}-2$ for every positive integer $s.$ Then, we claim
that the fourth cohomological transfer is an isomorphism in these internal
degrees. The content of the present Note subsection-wise can be described as
follows.

###  2.1 The degree $n_{s}:=5.2^{s}-2$

Notice that the Kameko homomorphism

$\begin{array}[]{ll}(\widetilde{Sq^{0}_{*}})_{n_{s}}:QP^{\otimes
4}_{n_{s}}&\longrightarrow QP^{\otimes 4}_{\frac{n_{s}-4}{2}}.\\\ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \mbox{[}\prod_{1\leq j\leq
4}t_{j}^{a_{j}}\mbox{]}&\longmapsto\left\\{\begin{array}[]{ll}\mbox{[}\prod_{1\leq
j\leq 4}t_{j}^{\frac{a_{j}-1}{2}}\mbox{]}&\text{if $a_{j}$ odd, for all
$j$},\\\ 0&\text{otherwise}\end{array}\right.\end{array}$

is an epimorphism of $\mathbb{Z}/2[GL_{4}]$-modules, hence we have an
isomorphism

$QP^{\otimes 4}_{n_{s}}\cong{\rm
Ker}((\widetilde{Sq^{0}_{*}})_{n_{s}})\bigoplus QP^{\otimes
4}_{\frac{n_{s}-4}{2}},$

which implies that

(5) $\dim(\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n_{s}}^{\otimes 4}]^{*})\leq\dim([{\rm
Ker}((\widetilde{Sq^{0}_{*}})_{n_{s}})]^{GL_{4}})+\dim(\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{\frac{n_{s}-4}{2}}^{\otimes 4}]^{*}).$

According to Lin [20], we deduce that

(6) ${\rm
Ext}_{\mathcal{A}}^{4,4+n_{s}}(\mathbb{Z}/2,\mathbb{Z}/2)=\left\\{\begin{array}[]{ll}\langle
h_{1}^{2}h_{2}^{2}\rangle=0&\mbox{if $s=1$},\\\\[2.84526pt] \langle
h_{1}^{3}h_{4},f_{0}\rangle=\langle h_{0}^{2}h_{2}h_{4},f_{0}\rangle&\mbox{if
$s=2$},\\\\[2.84526pt] \langle h_{0}^{2}h_{3}h_{5},e_{1}\rangle&\mbox{if
$s=3$},\\\\[2.84526pt] \langle h_{0}^{2}h_{s}h_{s+2}\rangle&\mbox{if $s\geq
4$}.\end{array}\right.$

By using these data and an admissible basis of ${\rm
Ker}((\widetilde{Sq^{0}_{*}})_{n_{s}})$ in [43, 44], we obtain the following
results.

The case $s=1$. We have

######  Theorem 2.1.1.

The coinvariant space $\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n_{1}}^{\otimes 4}]^{*}$ is trivial.

In order to prove the theorem, we use a basis of ${\rm
Ker}((\widetilde{Sq^{0}_{*}})_{n_{1}})$ and show that ${\rm
Ker}((\widetilde{Sq^{0}_{*}})_{n_{1}})$ is trivial. At the same time,
combining the inequality (5) and a theorem of Sum [46] that the coinvariant
space $(\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{\frac{n_{1}-4}{2}}^{\otimes 4}]^{*})$ is
trivial. Note that, this result has also been investigated in [8], in a
different way. Now, by Theorem 4.2.1 and the equality (6), it may be concluded
that

######  Corollary 2.1.2.

The rank 4 algebraic transfer

$Tr_{4}^{\mathcal{A}}:\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n_{1}}^{\otimes 4}]^{*}\to{\rm
Ext}_{\mathcal{A}}^{4,4+n_{1}}(\mathbb{Z}/2,\mathbb{Z}/2)$

is a trivial isomorphism.

Moreover, we have the following remarks.

######  Remark 2.1.3.

We consider the generic degree $12.2^{s}-4$ with $s$ an arbitrary positive
integer. It is easy to see that $\mu(12.2^{s}-4)=4$ for all $s>0.$ (Recall
that for each positive integer $n,$ by $\mu(n)$ one means the smallest number
$k$ for which it is possible to write $n=\sum_{1\leq i\leq k}(2^{d_{i}}-1),$
in which $d_{i}>0.$) Then, by Kameko’s Theorem [18], it follows that the
iterated Kameko map

$(\widetilde{Sq^{0}_{*}})^{s}_{12.2^{s}-4}:QP^{\otimes
4}_{12.2^{s}-4}\longrightarrow QP^{\otimes 4}_{8}$

is an isomorphism of $\mathbb{Z}/2GL_{4}$-modules, for any $s\geq 0.$
Combining this fact and Theorem 4.2.1, we conclude that the coinvariants
$\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{12.2^{s}-4}^{\otimes 4}]^{*}$ is trivial for
every non-negative integer $s.$ This also explained in [8], in another way. On
the other hand, by calculations of Lin [20], clearly ${\rm
Ext}_{\mathcal{A}}^{4,12.2^{s}}(\mathbb{Z}/2,\mathbb{Z}/2)=\mathbb{Z}/2.g_{s}$
for all $s>0.$ Therefore, the transfer homomorphism

$Tr_{4}^{\mathcal{A}}:\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{12.2^{s}-4}^{\otimes 4}]^{*}\to{\rm
Ext}_{\mathcal{A}}^{4,12.2^{s}}(\mathbb{Z}/2,\mathbb{Z}/2)$

is a monomorphism, but not an isomorphism for arbitrary $s>0.$

The case $s=2$. It is known, from a theorem of Sum [46], that
$\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{\frac{n_{2}-4}{2}}^{\otimes 4}]^{*}$ is
generated by $[x_{1}^{(0)}x_{2}^{(0)}x_{3}^{(0)}x_{4}^{(7)}].$ From this and a
direct computation using a basis of ${\rm
Ker}((\widetilde{Sq^{0}_{*}})_{n_{2}})$ and the inequality (5), we obtain

######  Theorem 2.1.4.

$\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n_{2}}^{\otimes 4}]^{*}$ is the vector space
of dimension $2$ with a basis consisting of 2 classes represented by
$x_{1}^{(1)}x_{2}^{(1)}x_{3}^{(1)}x_{4}^{(15)}$ and

$\begin{array}[]{ll}\zeta&=x_{1}^{(3)}x_{2}^{(5)}x_{3}^{(1)}x_{4}^{(9)}+x_{1}^{(3)}x_{2}^{(5)}x_{3}^{(2)}x_{4}^{(8)}+x_{1}^{(3)}x_{2}^{(6)}x_{3}^{(1)}x_{4}^{(8)}+x_{1}^{(3)}x_{2}^{(6)}x_{3}^{(2)}x_{4}^{(7)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(3)}x_{2}^{(5)}x_{3}^{(4)}x_{4}^{(6)}\\\
&\quad+x_{1}^{(3)}x_{2}^{(6)}x_{3}^{(3)}x_{4}^{(6)}+x_{1}^{(5)}x_{2}^{(6)}x_{3}^{(1)}x_{4}^{(6)}+x_{1}^{(3)}x_{2}^{(5)}x_{3}^{(5)}x_{4}^{(5)}+x_{1}^{(3)}x_{2}^{(6)}x_{3}^{(4)}x_{4}^{(5)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(5)}x_{2}^{(6)}x_{3}^{(2)}x_{4}^{(5)}\\\
&\quad+x_{1}^{(3)}x_{2}^{(9)}x_{3}^{(1)}x_{4}^{(5)}+x_{1}^{(5)}x_{2}^{(7)}x_{3}^{(1)}x_{4}^{(5)}+x_{1}^{(3)}x_{2}^{(9)}x_{3}^{(2)}x_{4}^{(4)}+x_{1}^{(5)}x_{2}^{(7)}x_{3}^{(2)}x_{4}^{(4)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(3)}x_{2}^{(10)}x_{3}^{(1)}x_{4}^{(4)}\\\
&\quad+x_{1}^{(6)}x_{2}^{(7)}x_{3}^{(1)}x_{4}^{(4)}+x_{1}^{(5)}x_{2}^{(6)}x_{3}^{(4)}x_{4}^{(3)}+x_{1}^{(6)}x_{2}^{(7)}x_{3}^{(2)}x_{4}^{(3)}+x_{1}^{(3)}x_{2}^{(10)}x_{3}^{(2)}x_{4}^{(3)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(3)}x_{2}^{(11)}x_{3}^{(2)}x_{4}^{(2)}\\\
&\quad+x_{1}^{(5)}x_{2}^{(9)}x_{3}^{(2)}x_{4}^{(2)}+x_{1}^{(6)}x_{2}^{(10)}x_{3}^{(1)}x_{4}^{(1)}+x_{1}^{(3)}x_{2}^{(5)}x_{3}^{(9)}x_{4}^{(1)}+x_{1}^{(3)}x_{2}^{(5)}x_{3}^{(8)}x_{4}^{(2)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(3)}x_{2}^{(6)}x_{3}^{(8)}x_{4}^{(1)}\\\
&\quad+x_{1}^{(3)}x_{2}^{(6)}x_{3}^{(7)}x_{4}^{(2)}+x_{1}^{(3)}x_{2}^{(5)}x_{3}^{(6)}x_{4}^{(4)}+x_{1}^{(3)}x_{2}^{(6)}x_{3}^{(6)}x_{4}^{(3)}+x_{1}^{(5)}x_{2}^{(6)}x_{3}^{(6)}x_{4}^{(1)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(3)}x_{2}^{(5)}x_{3}^{(5)}x_{4}^{(5)}\\\
&\quad+x_{1}^{(3)}x_{2}^{(6)}x_{3}^{(5)}x_{4}^{(4)}+x_{1}^{(5)}x_{2}^{(6)}x_{3}^{(5)}x_{4}^{(2)}+x_{1}^{(3)}x_{2}^{(9)}x_{3}^{(5)}x_{4}^{(1)}+x_{1}^{(5)}x_{2}^{(7)}x_{3}^{(5)}x_{4}^{(1)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(3)}x_{2}^{(9)}x_{3}^{(4)}x_{4}^{(2)}\\\
&\quad+x_{1}^{(5)}x_{2}^{(7)}x_{3}^{(4)}x_{4}^{(2)}+x_{1}^{(3)}x_{2}^{(10)}x_{3}^{(4)}x_{4}^{(1)}+x_{1}^{(6)}x_{2}^{(7)}x_{3}^{(4)}x_{4}^{(1)}+x_{1}^{(5)}x_{2}^{(6)}x_{3}^{(3)}x_{4}^{(4)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(6)}x_{2}^{(7)}x_{3}^{(3)}x_{4}^{(2)}\\\
&\quad+x_{1}^{(3)}x_{2}^{(10)}x_{3}^{(3)}x_{4}^{(2)}+x_{1}^{(3)}x_{2}^{(11)}x_{3}^{(2)}x_{4}^{(2)}+x_{1}^{(5)}x_{2}^{(9)}x_{3}^{(2)}x_{4}^{(2)}+x_{1}^{(6)}x_{2}^{(10)}x_{3}^{(1)}x_{4}^{(1)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(3)}x_{2}^{(12)}x_{3}^{(1)}x_{4}^{(2)}\\\
&\quad+x_{1}^{(7)}x_{2}^{(8)}x_{3}^{(1)}x_{4}^{(2)}+x_{1}^{(11)}x_{2}^{(4)}x_{3}^{(1)}x_{4}^{(2)}+x_{1}^{(13)}x_{2}^{(2)}x_{3}^{(1)}x_{4}^{(2)}+x_{1}^{(14)}x_{2}^{(1)}x_{3}^{(1)}x_{4}^{(2)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(12)}x_{2}^{(3)}x_{3}^{(1)}x_{4}^{(2)}\\\
&\quad+x_{1}^{(8)}x_{2}^{(7)}x_{3}^{(1)}x_{4}^{(2)}+x_{1}^{(4)}x_{2}^{(11)}x_{3}^{(1)}x_{4}^{(2)}+x_{1}^{(2)}x_{2}^{(13)}x_{3}^{(1)}x_{4}^{(2)}+x_{1}^{(1)}x_{2}^{(14)}x_{3}^{(1)}x_{4}^{(2)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(6)}x_{2}^{(6)}x_{3}^{(3)}x_{4}^{(3)}\\\
&\quad+x_{1}^{(5)}x_{2}^{(5)}x_{3}^{(5)}x_{4}^{(3)}+x_{1}^{(3)}x_{2}^{(3)}x_{3}^{(9)}x_{4}^{(3)}+x_{1}^{(5)}x_{2}^{(3)}x_{3}^{(7)}x_{4}^{(3)}+x_{1}^{(7)}x_{2}^{(7)}x_{3}^{(2)}x_{4}^{(2)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(6)}x_{2}^{(9)}x_{3}^{(1)}x_{4}^{(2)}\\\
&\quad+x_{1}^{(9)}x_{2}^{(6)}x_{3}^{(1)}x_{4}^{(2)}+x_{1}^{(10)}x_{2}^{(5)}x_{3}^{(1)}x_{4}^{(2)}+x_{1}^{(5)}x_{2}^{(10)}x_{3}^{(2)}x_{4}^{(1)}+x_{1}^{(13)}x_{2}^{(3)}x_{3}^{(1)}x_{4}^{(1)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(5)}x_{2}^{(11)}x_{3}^{(1)}x_{4}^{(1)}\\\
&\quad+x_{1}^{(9)}x_{2}^{(7)}x_{3}^{(1)}x_{4}^{(1)}.\end{array}$

To check that $\zeta_{2}\in{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n_{2}}^{\otimes 4}]^{*},$ we need only to
consider the effects of the Steenrod squares $Sq^{i}$ for $i=1,2,4,8$ because
of the unstable condition. Now, noticing that $\lambda_{1}\in\Lambda^{1,1},$
$\lambda_{15}\in\Lambda^{1,15}$ and
$\widetilde{f}_{0}=\lambda_{4}\lambda_{6}\lambda_{5}\lambda_{3}+\lambda_{5}\lambda_{7}\lambda^{2}_{3}+\lambda_{3}^{2}\lambda_{2}\lambda_{5}\lambda_{7}+\lambda_{2}\lambda_{4}\lambda_{5}\lambda_{7}\in\Lambda^{4,n_{2}}$
are the cycles in the algebra $\Lambda.$ Moreover, they are representative of
the non-zero elements $h_{i}\in{\rm
Ext}_{\mathcal{A}}^{1,2^{i}}(\mathbb{Z}/2,\mathbb{Z}/2),$ for $i=1,4$ and
$f_{0}\in{\rm Ext}_{\mathcal{A}}^{4,4+n_{2}}(\mathbb{Z}/2,\mathbb{Z}/2)$
respectively. Then, using Theorems 2.1.4 and the representation of
$Tr_{4}^{\mathcal{A}}$ over $\Lambda,$ we see that the cycles
$\lambda_{1}^{3}\lambda_{15}=\psi_{4}(x_{1}^{(1)}x_{2}^{(1)}x_{3}^{(1)}x_{4}^{(15)})$
and
$\widetilde{f}_{0}+\delta(\lambda_{3}\lambda_{5}\lambda_{11})=\psi_{4}(\zeta)$
in the lambda algebra are representative of the non-zero elements
$h_{0}^{2}h_{2}h_{4}$ and $f_{0}$ respectively. This shows that
$h_{1}^{3}h_{4}$ and $f_{0}$ are in the image of $Tr_{4}^{\mathcal{A}}.$
Combining this with Theorem 2.1.4 and the equality (6), it follows that

######  Corollary 2.1.5.

The fourth cohomological transfer is a isomorphism in the internal degree
$n_{2}.$

The case $s=3$. We consider the element
$\widetilde{\zeta}\in[P_{n_{3}}^{\otimes 4}]^{*},$ which is the following sum:

$\begin{array}[]{ll}&x_{1}^{(11)}x_{2}^{(11)}x_{3}^{(11)}x_{4}^{(5)}+x_{1}^{(11)}x_{2}^{(11)}x_{3}^{(13)}x_{4}^{(3)}+x_{1}^{(7)}x_{2}^{(11)}x_{3}^{(17)}x_{4}^{(3)}+x_{1}^{(11)}x_{2}^{(7)}x_{3}^{(17)}x_{4}^{(3)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(7)}x_{2}^{(13)}x_{3}^{(15)}x_{4}^{(3)}\\\
&+x_{1}^{(11)}x_{2}^{(15)}x_{3}^{(9)}x_{4}^{(3)}+x_{1}^{(15)}x_{2}^{(11)}x_{3}^{(9)}x_{4}^{(3)}+x_{1}^{(7)}x_{2}^{(19)}x_{3}^{(9)}x_{4}^{(3)}+x_{1}^{(19)}x_{2}^{(7)}x_{3}^{(9)}x_{4}^{(3)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(7)}x_{2}^{(19)}x_{3}^{(7)}x_{4}^{(5)}\\\
&+x_{1}^{(19)}x_{2}^{(7)}x_{3}^{(7)}x_{4}^{(5)}+x_{1}^{(11)}x_{2}^{(19)}x_{3}^{(5)}x_{4}^{(3)}+x_{1}^{(19)}x_{2}^{(11)}x_{3}^{(5)}x_{4}^{(3)}+x_{1}^{(11)}x_{2}^{(21)}x_{3}^{(3)}x_{4}^{(3)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(19)}x_{2}^{(13)}x_{3}^{(3)}x_{4}^{(3)}\\\
&+x_{1}^{(7)}x_{2}^{(23)}x_{3}^{(5)}x_{4}^{(3)}+x_{1}^{(23)}x_{2}^{(7)}x_{3}^{(5)}x_{4}^{(3)}+x_{1}^{(11)}x_{2}^{(11)}x_{3}^{(7)}x_{4}^{(9)}+x_{1}^{(11)}x_{2}^{(7)}x_{3}^{(11)}x_{4}^{(9)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(7)}x_{2}^{(11)}x_{3}^{(11)}x_{4}^{(9)}\\\
&+x_{1}^{(7)}x_{2}^{(25)}x_{3}^{(3)}x_{4}^{(3)}+x_{1}^{(23)}x_{2}^{(9)}x_{3}^{(3)}x_{4}^{(3)}+x_{1}^{(15)}x_{2}^{(17)}x_{3}^{(3)}x_{4}^{(3)}+x_{1}^{(15)}x_{2}^{(15)}x_{3}^{(3)}x_{4}^{(5)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(27)}x_{2}^{(5)}x_{3}^{(3)}x_{4}^{(3)}\\\
&+x_{1}^{(29)}x_{2}^{(3)}x_{3}^{(3)}x_{4}^{(3)}+x_{1}^{(13)}x_{2}^{(11)}x_{3}^{(7)}x_{4}^{(7)}+x_{1}^{(11)}x_{2}^{(7)}x_{3}^{(13)}x_{4}^{(7)}+x_{1}^{(7)}x_{2}^{(13)}x_{3}^{(11)}x_{4}^{(7)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(13)}x_{2}^{(7)}x_{3}^{(7)}x_{4}^{(11)}\\\
&+x_{1}^{(7)}x_{2}^{(7)}x_{3}^{(13)}x_{4}^{(11)}+x_{1}^{(7)}x_{2}^{(13)}x_{3}^{(7)}x_{4}^{(11)}+x_{1}^{(11)}x_{2}^{(7)}x_{3}^{(7)}x_{4}^{(13)}+x_{1}^{(7)}x_{2}^{(11)}x_{3}^{(7)}x_{4}^{(13)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(7)}x_{2}^{(7)}x_{3}^{(11)}x_{4}^{(13)}\\\
&+x_{1}^{(7)}x_{2}^{(7)}x_{3}^{(7)}x_{4}^{(17)}+x_{1}^{(7)}x_{2}^{(7)}x_{3}^{(9)}x_{4}^{(15)}+x_{1}^{(7)}x_{2}^{(11)}x_{3}^{(5)}x_{4}^{(15)}+x_{1}^{(7)}x_{2}^{(13)}x_{3}^{(3)}x_{4}^{(15)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(7)}x_{2}^{(7)}x_{3}^{(19)}x_{4}^{(5)}\\\
&+x_{1}^{(7)}x_{2}^{(7)}x_{3}^{(21)}x_{4}^{(3)}+x_{1}^{(11)}x_{2}^{(7)}x_{3}^{(15)}x_{4}^{(5)}+x_{1}^{(11)}x_{2}^{(15)}x_{3}^{(7)}x_{4}^{(5)}+x_{1}^{(15)}x_{2}^{(11)}x_{3}^{(7)}x_{4}^{(5)}.\end{array}$

In [38], we showed that $(\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{\frac{n_{3}-4}{2}}^{\otimes 4}]^{*})$ is
1-dimensional. Basing this coupled with the inequality (5) and an admissible
basis of ${\rm Ker}((\widetilde{Sq^{0}_{*}})_{n_{3}}),$ it may be claimed that

######  Theorem 2.1.6.

The space of $GL_{4}$-coinvariants $\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n_{3}}^{\otimes 4}]^{*}$ has dimension $2$
with the basis
$\\{[x_{1}^{(0)}x_{2}^{(0)}x_{3}^{(7)}x_{4}^{(31)}],[\widetilde{\zeta}]\\}.$

By the unstable condition, to verify that $\widetilde{\zeta}\in{\rm
Ext}_{\mathcal{A}}^{0,n_{3}}(\mathbb{Z}/2,P^{\otimes 4}),$ we need only to
compute the actions of the Steenrod squares $Sq^{2^{i}}$ for $i=0,1,2,3.$ Now,
clearly

$\overline{e}_{1}=\lambda_{7}^{3}\lambda_{17}+(\lambda_{7}\lambda_{11}^{2}+\lambda_{7}^{2}\lambda_{15})\lambda_{9}+\lambda_{15}\lambda_{11}\lambda_{7}\lambda_{5}+\lambda_{7}^{2}\lambda_{11}\lambda_{13}\in\Lambda^{4,n_{3}}$

is a cycle in $\Lambda$ and $e_{1}=[\overline{e}_{1}]\in{\rm
Ext}_{\mathcal{A}}^{4,4+n_{3}}(\mathbb{Z}/2,\mathbb{Z}/2)$. Then, based on
Theorems 2.1.6 and the representation in the lambda algebra of the rank 4
transfer, we claim that the cycles

$\lambda_{0}^{2}\lambda_{7}\lambda_{31}=\psi_{4}(x_{1}^{(0)}x_{2}^{(0)}x_{3}^{(7)}x_{4}^{(31)}),\
\ \overline{e}_{1}=\psi_{4}(\widetilde{\zeta})$

in $\Lambda^{4,n_{3}}$ are representative of the non-zero elements
$h_{0}^{2}h_{3}h_{5}$ and $e_{1}$ respectively. This shows that
$h_{0}^{2}h_{3}h_{5}$ and $e_{1}$ are in the image of $Tr_{4}^{\mathcal{A}}$
and therefore by Theorem 2.1.4 and the equality (6), it follows that

######  Corollary 2.1.7.

The Singer transfer

$Tr_{4}^{\mathcal{A}}:\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n_{3}}^{\otimes 4}]^{*}\to{\rm
Ext}_{\mathcal{A}}^{4,4+n_{3}}(\mathbb{Z}/2,\mathbb{Z}/2)$

is an isomorphism.

The case $s\geq 4$. Direct calculating based on an admissible basis of ${\rm
Ker}((\widetilde{Sq^{0}_{*}})_{n_{s}}),$ we get

######  Theorem 2.1.8.

For each $s\geq 4,$ the space $\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n_{s}}^{\otimes 4}]^{*}$ is one-dimensional
and generated by
$[x_{1}^{(0)}x_{2}^{(0)}x_{3}^{(2^{s}-1)}x_{4}^{(2^{s+2}-1)}].$

Using the representation of $Tr_{4}^{\mathcal{A}}$ over $\Lambda$, we conclude
that the cycles

$\lambda_{0}^{2}\lambda_{2^{s}-1}\lambda_{2^{s+2}-1}=\psi_{4}(x_{1}^{(0)}x_{2}^{(0)}x_{3}^{(2^{s}-1)}x_{4}^{(2^{s+2}-1)})$

in $\Lambda^{4,n_{s}}$ are representative of the non-zero elements
$h_{0}^{2}h_{s}h_{s+2}\in{\rm
Ext}_{\mathcal{A}}^{4,4+n_{s}}(\mathbb{Z}/2,\mathbb{Z}/2)$ for all $s\geq 4.$
This leads to $h_{0}^{2}h_{s}h_{s+2}$ being in the image of
$Tr_{4}^{\mathcal{A}}.$ So, combining these data with the equality (6) and
Theorem 2.1.8, the following is immediate.

######  Corollary 2.1.9.

The cohomological transfers
$Tr_{4}^{\mathcal{A}}:\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n_{s}}^{\otimes 4}]^{*}\to{\rm
Ext}_{\mathcal{A}}^{4,4+n_{s}}(\mathbb{Z}/2,\mathbb{Z}/2)$ are isomorphisms,
for all $s\geq 4.$

###  2.2 The degree $n^{\prime}_{s}:=17.2^{s}-2$

Because Kameko’s squaring operation
$(\widetilde{Sq^{0}_{*}})_{n^{\prime}_{s}}:QP^{\otimes
4}_{n^{\prime}_{s}}\longrightarrow QP^{\otimes
4}_{\frac{n^{\prime}_{s}-4}{2}}$ is an epimorphism of
$\mathbb{Z}/2[GL_{4}]$-modules, we have

(7) $\dim(\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n^{\prime}_{s}}^{\otimes
4}]^{*})\leq\dim([{\rm
Ker}((\widetilde{Sq^{0}_{*}})_{n^{\prime}_{s}})]^{GL_{4}})+\dim(\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{\frac{n^{\prime}_{s}-4}{2}}^{\otimes
4}]^{*}).$

By direct calculations using a monomial basis of ${\rm
Ker}((\widetilde{Sq^{0}_{*}})_{n^{\prime}_{s}})$ in Sum [43, 44], we get the
following results.

The case $s=1$. Consider the following element in $[P^{\otimes
4}_{n^{\prime}_{1}}]^{*}$:

$\begin{array}[]{ll}\overline{\zeta}&=x_{1}^{(3)}x_{2}^{(13)}x_{3}^{(7)}x_{4}^{(9)}+x_{1}^{(3)}x_{2}^{(13)}x_{3}^{(11)}x_{4}^{(5)}+x_{1}^{(3)}x_{2}^{(13)}x_{3}^{(13)}x_{4}^{(3)}+x_{1}^{(5)}x_{2}^{(11)}x_{3}^{(7)}x_{4}^{(9)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(5)}x_{2}^{(11)}x_{3}^{(11)}x_{4}^{(5)}\\\
&+x_{1}^{(5)}x_{2}^{(11)}x_{3}^{(13)}x_{4}^{(3)}+x_{1}^{(5)}x_{2}^{(13)}x_{3}^{(7)}x_{4}^{(7)}+x_{1}^{(7)}x_{2}^{(3)}x_{3}^{(11)}x_{4}^{(11)}+x_{1}^{(7)}x_{2}^{(3)}x_{3}^{(13)}x_{4}^{(9)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(7)}x_{2}^{(5)}x_{3}^{(11)}x_{4}^{(9)}\\\
&+x_{1}^{(7)}x_{2}^{(5)}x_{3}^{(13)}x_{4}^{(7)}+x_{1}^{(7)}x_{2}^{(7)}x_{3}^{(7)}x_{4}^{(11)}+x_{1}^{(7)}x_{2}^{(7)}x_{3}^{(9)}x_{4}^{(9)}+x_{1}^{(7)}x_{2}^{(7)}x_{3}^{(13)}x_{4}^{(5)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(7)}x_{2}^{(9)}x_{3}^{(7)}x_{4}^{(9)}\\\
&+x_{1}^{(7)}x_{2}^{(11)}x_{3}^{(5)}x_{4}^{(9)}+x_{1}^{(7)}x_{2}^{(13)}x_{3}^{(3)}x_{4}^{(9)}+x_{1}^{(7)}x_{2}^{(13)}x_{3}^{(5)}x_{4}^{(7)}+x_{1}^{(9)}x_{2}^{(7)}x_{3}^{(7)}x_{4}^{(9)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(9)}x_{2}^{(7)}x_{3}^{(11)}x_{4}^{(5)}\\\
&+x_{1}^{(9)}x_{2}^{(7)}x_{3}^{(13)}x_{4}^{(3)}+x_{1}^{(11)}x_{2}^{(3)}x_{3}^{(7)}x_{4}^{(11)}+x_{1}^{(11)}x_{2}^{(3)}x_{3}^{(13)}x_{4}^{(5)}+x_{1}^{(11)}x_{2}^{(5)}x_{3}^{(11)}x_{4}^{(5)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(11)}x_{2}^{(7)}x_{3}^{(3)}x_{4}^{(11)}\\\
&+x_{1}^{(11)}x_{2}^{(7)}x_{3}^{(9)}x_{4}^{(5)}+x_{1}^{(11)}x_{2}^{(9)}x_{3}^{(7)}x_{4}^{(5)}+x_{1}^{(11)}x_{2}^{(11)}x_{3}^{(3)}x_{4}^{(7)}+x_{1}^{(11)}x_{2}^{(11)}x_{3}^{(5)}x_{4}^{(5)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(11)}x_{2}^{(13)}x_{3}^{(3)}x_{4}^{(5)}\\\
&+x_{1}^{(13)}x_{2}^{(3)}x_{3}^{(13)}x_{4}^{(3)}+x_{1}^{(13)}x_{2}^{(5)}x_{3}^{(11)}x_{4}^{(3)}+x_{1}^{(13)}x_{2}^{(7)}x_{3}^{(7)}x_{4}^{(5)}+x_{1}^{(13)}x_{2}^{(7)}x_{3}^{(7)}x_{4}^{(5)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(13)}x_{2}^{(7)}x_{3}^{(9)}x_{4}^{(3)}\\\
&+x_{1}^{(13)}x_{2}^{(9)}x_{3}^{(7)}x_{4}^{(3)}+x_{1}^{(13)}x_{2}^{(11)}x_{3}^{(5)}x_{4}^{(3)}+x_{1}^{(13)}x_{2}^{(13)}x_{3}^{(3)}x_{4}^{(3)}.\end{array}$

######  Theorem 2.2.1.

The space $\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n^{\prime}_{1}}^{\otimes 4}]^{*}$ is
1-dimensional and generated by $[\overline{\zeta}].$

An effective approach, based on the inequality (7) coupled with a result of
Sum [46] and the calculations of the invariant space $[{\rm
Ker}_{n^{\prime}_{1}}]^{GL_{4}}$, is presented in the proof of the theorem.
Now, it is easy to see that
$\overline{d}_{1}=\lambda_{7}^{2}\lambda_{5}\lambda_{13}+\lambda_{7}^{2}\lambda_{9}^{2}+\lambda_{7}\lambda_{11}\lambda_{9}\lambda_{5}+\lambda_{15}\lambda_{3}\lambda_{11}\lambda_{3}\in\Lambda^{4,n^{\prime}_{1}}$
is a cycle in $\Lambda$ and is a representative of the non-zero element
$d_{1}\in{\rm
Ext}_{\mathcal{A}}^{4,4+n^{\prime}_{1}}(\mathbb{Z}/2,\mathbb{Z}/2).$ Then,
using Theorem 2.2.1 and a representation in $\Lambda$ of
$Tr_{4}^{\mathcal{A}},$ we deduce that
$d_{1}=Tr_{4}([\overline{\zeta}])=[\psi_{4}(\overline{\zeta})].$ This implies
that $d_{1}$ is in the image of $Tr_{4}^{\mathcal{A}}.$ Combining this with
Theorem 2.2.1 and the fact that ${\rm
Ext}_{\mathcal{A}}^{4,4+n^{\prime}_{1}}(\mathbb{Z}/2,\mathbb{Z}/2)$ has
dimension one, we immediately obtain

######  Corollary 2.2.2.

The transfer $Tr_{4}^{\mathcal{A}}$ is an isomorphism when acting on
$\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n^{\prime}_{1}}^{\otimes 4}]^{*}.$

The case $s=2$. Using the inequality (7) and a result in Sum [46], we have

######  Theorem 2.2.3.

The coinvariant space $\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n^{\prime}_{2}}^{\otimes 4}]^{*}$ is the
$\mathbb{Z}/2$-vector space of 1 dimension with the basis
$\\{[x_{1}^{(1)}x_{2}^{(1)}x_{3}^{(1)}x_{4}^{(63)}]\\}.$

We observe that $\lambda_{1}\in\Lambda^{1,1}$ and
$\lambda_{63}\in\Lambda^{1,63}$ are the cycles in $\Lambda$ and
$[\lambda_{1}]=h_{1}\in{\rm
Ext}_{\mathcal{A}}^{1,2}(\mathbb{Z}/2,\mathbb{Z}/2)$ and
$[\lambda_{63}]=h_{6}\in{\rm
Ext}_{\mathcal{A}}^{1,64}(\mathbb{Z}/2,\mathbb{Z}/2).$ So, from Theorem 2.2.3
and the representation of the fourth transfer homomorphism over the algebra
$\Lambda,$ we claim that the cycle
$\lambda_{1}^{3}\lambda_{63}=\psi_{4}(x_{1}^{(1)}x_{2}^{(1)}x_{3}^{(1)}x_{4}^{(63)})$
in $\Lambda$ is a representative of the non-zero element
$h_{1}^{3}h_{6}=h_{0}^{2}h_{2}h_{6}\in{\rm
Ext}_{\mathcal{A}}^{4,4+n^{\prime}_{2}}(\mathbb{Z}/2,\mathbb{Z}/2).$ This
shows that $h_{1}^{3}h_{6}\in{\rm Im}(Tr_{4}^{\mathcal{A}})$ and so, the
following corollary is immediate from the fact that ${\rm
Ext}_{\mathcal{A}}^{4,4+n^{\prime}_{2}}(\mathbb{Z}/2,\mathbb{Z}/2)$ is
1-dimensional.

######  Corollary 2.2.4.

The algebraic transfer $Tr_{4}^{\mathcal{A}}:\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n^{\prime}_{2}}^{\otimes 4}]^{*}\to{\rm
Ext}_{\mathcal{A}}^{4,4+n^{\prime}_{2}}(\mathbb{Z}/2,\mathbb{Z}/2)$ is an
isomorphism.

The case $s\geq 3$. The following theorem is proved by using the equality (7)
and our result in [38].

######  Theorem 2.2.5.

Let $s$ be a positive integer such that $s\geq 3.$ Then, we have

$\dim\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n^{\prime}_{s}}^{\otimes
4}]^{*}=\left\\{\begin{array}[]{ll}1&\mbox{if $s=4$},\\\ 2&\mbox{if $s\neq
4$}.\end{array}\right.$

Furthermore,

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt&\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n^{\prime}_{s}}^{\otimes 4}]^{*}\\\
&=\left\\{\begin{array}[]{ll}\langle[x_{1}^{(1)}x_{2}^{(7)}x_{3}^{(63)}x_{4}^{(63)}],[x_{1}^{(0)}x_{2}^{(0)}x_{3}^{(7)}x_{4}^{(127)}]\rangle&\mbox{if
$s=3$},\\\\[2.84526pt]
\langle[x_{1}^{(1)}x_{2}^{(15)}x_{3}^{(127)}x_{4}^{(127)}]\rangle&\mbox{if
$s=4$},\\\\[2.84526pt]
\langle[x_{1}^{(1)}x_{2}^{(2^{s-1}-1)}x_{3}^{(2^{s-1}-1)}x_{4}^{(2^{s+4}-1)}],[x_{1}^{(1)}x_{2}^{(2^{s}-1)}x_{3}^{(2^{s+3}-1)}x_{4}^{(2^{s+3}-1)}]\rangle&\mbox{if
$s\geq 5$}.\end{array}\right.\end{array}$

From this theorem and the representation in $\Lambda$ of the rank 4 transfer,
it may be concluded that

$\begin{array}[]{ll}&[\psi_{4}(x_{1}^{(1)}x_{2}^{(7)}x_{3}^{(63)}x_{4}^{(63)})]=[\lambda_{1}\lambda_{7}\lambda_{63}^{2}]\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&=Tr_{4}^{\mathcal{A}}([x_{1}^{(1)}x_{2}^{(7)}x_{3}^{(63)}x_{4}^{(63)}])=h_{1}h_{3}h_{6}^{2}\in{\rm
Ext}_{\mathcal{A}}^{4,4+n^{\prime}_{3}}(\mathbb{Z}/2,\mathbb{Z}/2),\\\
&[\psi_{4}(x_{1}^{(0)}x_{2}^{(0)}x_{3}^{(7)}x_{4}^{(127)})]=[\lambda_{0}\lambda_{7}\lambda_{127}^{2}]\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&=Tr_{4}^{\mathcal{A}}([x_{1}^{(0)}x_{2}^{(0)}x_{3}^{(7)}x_{4}^{(127)}])=h_{0}^{2}h_{3}h_{7}\in{\rm
Ext}_{\mathcal{A}}^{4,4+n^{\prime}_{3}}(\mathbb{Z}/2,\mathbb{Z}/2),\\\
&[\psi_{4}(x_{1}^{(1)}x_{2}^{(15)}x_{3}^{(127)}x_{4}^{(127)})]=[\lambda_{1}\lambda_{15}\lambda_{127}^{2}]\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&=Tr_{4}^{\mathcal{A}}([x_{1}^{(1)}x_{2}^{(15)}x_{3}^{(127)}x_{4}^{(127)}])=h_{1}h_{4}h_{7}^{3}\in{\rm
Ext}_{\mathcal{A}}^{4,4+n^{\prime}_{4}}(\mathbb{Z}/2,\mathbb{Z}/2),\\\
&[\psi_{4}(x_{1}^{(1)}x_{2}^{(2^{s-1}-1)}x_{3}^{(2^{s-1}-1)}x_{4}^{(2^{s+4}-1)})]=[\lambda_{1}\lambda^{2}_{2^{s-1}-1}\lambda_{2^{s+4}-1}]\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&=Tr_{4}^{\mathcal{A}}([x_{1}^{(1)}x_{2}^{(2^{s-1}-1)}x_{3}^{(2^{s-1}-1)}x_{4}^{(2^{s+4}-1)}])=h_{1}h_{s-1}^{2}h_{s+4}\in{\rm
Ext}_{\mathcal{A}}^{4,4+n^{\prime}_{s}}(\mathbb{Z}/2,\mathbb{Z}/2)\ \mbox{for
$s\geq 5$}\\\
&[\psi_{4}(x_{1}^{(1)}x_{2}^{(2^{s}-1)}x_{3}^{(2^{s+3}-1)}x_{4}^{(2^{s+3}-1)})]=[\lambda_{1}\lambda_{2^{s}-1}\lambda_{2^{s+3}-1}^{2}]\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&=Tr_{4}^{\mathcal{A}}([x_{1}^{(1)}x_{2}^{(2^{s}-1)}x_{3}^{(2^{s+3}-1)}x_{4}^{(2^{s+3}-1)}])=h_{1}h_{s}h_{s+3}^{2}\in{\rm
Ext}_{\mathcal{A}}^{4,4+n^{\prime}_{s}}(\mathbb{Z}/2,\mathbb{Z}/2)\ \mbox{for
$s\geq 5$}.\end{array}$

Combining these with the fact that

$\dim{\rm
Ext}_{\mathcal{A}}^{4,4+n^{\prime}_{s}}(\mathbb{Z}/2,\mathbb{Z}/2)=\left\\{\begin{array}[]{ll}1&\mbox{if
$s=4$},\\\ 2&\mbox{if $s=3$ and $s\geq 5$},\end{array}\right.$

we have immediately

######  Corollary 2.2.6.

For each $s>2,$ the cohomological transfer

$Tr_{4}^{\mathcal{A}}:\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n^{\prime}_{s}}^{\otimes 4}]^{*}\to{\rm
Ext}_{\mathcal{A}}^{4,4+n^{\prime}_{s}}(\mathbb{Z}/2,\mathbb{Z}/2)$

is also an isomorphism.

Thus, Corollaries 2.1.2 \- 2.2.6 confirmed Singer’s conjecture that
$Tr_{4}^{\mathcal{A}}$ is a monomorphism in all the internal degrees
$q.2^{s}-2$ for $q\in\\{5,17\\}$ and all positive integers $s.$ Moreover,
motivated by the data in this paper, we propose the following.

######  Conjecture 2.2.7.

Let us considere degrees $2^{s+m}+2^{s}-2$, where $s,\,m$ are positive
integers. Then, Singer’s conjecture for the rank 4 transfer holds in these
internal degrees.

So, by our previous results [38, 40] and the work of Sum [46], we see that if
this conjecture is true then the fourth algebraic transfer is a monomorphism
for all degrees $n.$ This means that Singer’s conjecture is true for
cohomological degrees less than or equal to 4.

##  3 Proofs of main results

We first provide some definitions, which will be used in proofs of our main
results.

######  Definition 3.1.

Let us consider the polynomials $u$ and $v$ in $P^{\otimes h}_{n}.$ We say
that $u\equiv v$ if and only if $(u+v)$ is $\mathcal{A}$-decomposable (or
”hit”).

The readers can see that the binary relation ”$\equiv$” on $P^{\otimes h}_{n}$
is an equivalence relation.

######  Definition 3.2.

For each $1\leq j\leq h,$ we define the $\mathbb{Z}/2$-linear map

$\theta_{j}:V_{h}\cong\langle t_{1},\ldots,t_{h}\rangle\to V_{h}\cong\langle
t_{1},\ldots,t_{h}\rangle$

by subsituting $\theta_{j}(t_{j})=t_{j+1},$ $\theta_{j}(t_{j+1})=t_{j},$
$\theta_{j}(t_{i})=t_{i},$ for $i\neq j,\,j+1,$ $1\leq j<4,$ and
$\theta_{h}(t_{1})=t_{1}+t_{2},$ $\theta_{4}(t_{i})=t_{i},$ for $1<i\leq h.$

From this definition, the group $GL_{h}\cong GL(V_{h})$ is generated by
$\theta_{j},$ for $1\leq j\leq h,$ and the symmetric group $S_{h}$ is
generated by $\theta_{j}$ for $1\leq j<h.$ Further, $\theta_{j}$ induces a
homomorphism of $\mathcal{A}$-algebras which is also denoted by
$\theta_{j}:P^{\otimes h}\to P^{\otimes h}.$ This together with Definition 3.1
say that a class $[u]\in QP_{n}^{\otimes h}$ is an $S_{h}$-invariant (resp.
$GL_{h}$-invariant) if and only if $\theta_{j}(u)\equiv u$ for $1\leq j<h$
(resp. $1\leq j\leq h$).

Now, in what follows: for any admissible monomials ${\rm adm}_{1},\ldots,{\rm
adm}_{k}$ in $P^{\otimes 4}_{n}$ and for a subgroup $\mathscr{G}$ of $GL_{4},$
let us denote $\mathscr{G}({\rm adm}_{1},\ldots,{\rm adm}_{k})$ the
$\mathbb{Z}/2\mathscr{G}$-submodule of ${\rm Ker}_{n}:={\rm
Ker}((\widetilde{Sq^{0}_{*}})_{n})$ generated by the set $\\{[{\rm
adm}_{j}]:\,1\leq j\leq k\\}.$

We have an isomorphism

${\rm Ker}_{n}\cong\underline{{\rm Ker}_{n}}\bigoplus\widehat{{\rm Ker}_{n}},$

where

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus 2.0pt\underline{{\rm
Ker}_{n}}&:=\langle\\{[t_{1}^{a_{1}}t_{2}^{a_{2}}t_{3}^{a_{3}}t_{4}^{a_{4}}]\in{\rm
Ker}_{n}:\,a_{1}a_{2}a_{3}a_{4}=0\\}\rangle,\\\ \widehat{{\rm
Ker}_{n}}&:=\langle\\{[t_{1}^{a_{1}}t_{2}^{a_{2}}t_{3}^{a_{3}}t_{4}^{a_{4}}]\in{\rm
Ker}_{n}:\,a_{1}a_{2}a_{3}a_{4}\neq 0\\}\rangle\end{array}$

are $\mathbb{Z}/2$-subspaces of ${\rm Ker}_{n}.$

###  3.1 The degree $n_{s}=5.2^{s}-2$

In order to prove Theorems 4.2.1, 2.1.4, 2.1.6 and 2.1.8, we need to some
related results.

The case $s=1$. According to Sum [43, 44], $\underline{{\rm Ker}_{n_{1}}}$ has
a basis consisting of all the classes represented by the admissible monomials
${\rm adm}_{1,\,j}$ for all $1\leq j\leq 42,$ where

${\rm adm}_{1,\,1}=t_{3}t_{4}^{7}$, |  ${\rm adm}_{1,\,2}=t_{3}^{7}t_{4}$, |  ${\rm adm}_{1,\,3}=t_{2}t_{4}^{7}$, |  ${\rm adm}_{1,\,4}=t_{2}t_{3}^{7}$,
---|---|---|---
${\rm adm}_{1,\,5}=t_{2}^{7}t_{4}$, |  ${\rm adm}_{1,\,6}=t_{2}^{7}t_{3}$, |  ${\rm adm}_{1,\,7}=t_{1}t_{4}^{7}$, |  ${\rm adm}_{1,\,8}=t_{1}t_{3}^{7}$,
${\rm adm}_{1,\,9}=t_{1}t_{2}^{7}$, |  ${\rm adm}_{1,\,10}=t_{1}^{7}t_{4}$, |  ${\rm adm}_{1,\,11}=t_{1}^{7}t_{3}$, |  ${\rm adm}_{1,\,12}=t_{1}^{7}t_{2}$,
${\rm adm}_{1,\,13}=t_{3}^{3}t_{4}^{5}$, |  ${\rm adm}_{1,\,14}=t_{2}^{3}t_{4}^{5}$, |  ${\rm adm}_{1,\,15}=t_{2}^{3}t_{3}^{5}$, |  ${\rm adm}_{1,\,16}=t_{1}^{3}t_{4}^{5}$,
${\rm adm}_{1,\,17}=t_{1}^{3}t_{3}^{5}$, |  ${\rm adm}_{1,\,18}=t_{1}^{3}t_{2}^{5}$, |  ${\rm adm}_{1,\,19}=t_{2}t_{3}t_{4}^{6}$, |  ${\rm adm}_{1,\,20}=t_{2}t_{3}^{6}t_{4}$,

${\rm adm}_{1,\,21}=t_{1}t_{3}t_{4}^{6}$, |  ${\rm adm}_{1,\,22}=t_{1}t_{3}^{6}t_{4}$, |  ${\rm adm}_{1,\,23}=t_{1}t_{2}t_{4}^{6}$, |  ${\rm adm}_{1,\,24}=t_{1}t_{2}t_{3}^{6}$,
---|---|---|---
${\rm adm}_{1,\,25}=t_{1}t_{2}^{6}t_{4}$, |  ${\rm adm}_{1,\,26}=t_{1}t_{2}^{6}t_{3}$, |  ${\rm adm}_{1,\,27}=t_{2}t_{3}^{2}t_{4}^{5}$, |  ${\rm adm}_{1,\,28}=t_{1}t_{3}^{2}t_{4}^{5}$,
${\rm adm}_{1,\,29}=t_{1}t_{2}^{2}t_{4}^{5}$, |  ${\rm adm}_{1,\,30}=t_{1}t_{2}^{2}t_{3}^{5}$, |  ${\rm adm}_{1,\,31}=t_{2}t_{3}^{3}t_{4}^{4}$, |  ${\rm adm}_{1,\,32}=t_{2}^{3}t_{3}t_{4}^{4}$,
${\rm adm}_{1,\,33}=t_{2}^{3}t_{3}^{4}t_{4}$, |  ${\rm adm}_{1,\,34}=t_{1}t_{3}^{3}t_{4}^{4}$, |  ${\rm adm}_{1,\,35}=t_{1}t_{2}^{3}t_{4}^{4}$, |  ${\rm adm}_{1,\,36}=t_{1}t_{2}^{3}t_{3}^{4}$,
${\rm adm}_{1,\,37}=t_{1}^{3}t_{3}t_{4}^{4}$, |  ${\rm adm}_{1,\,38}=t_{1}^{3}t_{2}t_{4}^{4}$, |  ${\rm adm}_{1,\,39}=t_{1}^{3}t_{2}t_{3}^{4}$, |  ${\rm adm}_{1,\,40}=t_{1}^{3}t_{3}^{4}t_{4}$,
${\rm adm}_{1,\,41}=t_{1}^{3}t_{2}^{4}t_{4}$, |  ${\rm adm}_{1,\,42}=t_{1}^{3}t_{2}^{4}t_{3}$. |  |

From the admissible basis above, by a simple computation, we see that

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus 2.0pt&S_{4}({\rm
adm}_{1,\,1})=\langle\\{[{\rm adm}_{1,\,j}]:\,1\leq j\leq 12\\}\rangle,\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt&S_{4}({\rm adm}_{1,\,13})=\langle\\{[{\rm
adm}_{1,\,j}]:\,13\leq j\leq 18]\\}\rangle,\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt&S_{4}({\rm adm}_{1,\,19},\ {\rm adm}_{1,\,27})=\langle\\{[{\rm
adm}_{1,\,j}]:\,19\leq j\leq 42\\}\rangle\\\ \end{array}$

are $S_{4}$-submodules of $\underline{{\rm Ker}_{n_{1}}}.$ Therefore, we have
an isomorphism

$\underline{{\rm Ker}_{n_{1}}}\cong S_{4}({\rm adm}_{1,\,1})\bigoplus
S_{4}({\rm adm}_{1,\,13})\bigoplus S_{4}({\rm adm}_{1,\,19},\ {\rm
adm}_{1,\,27}).$

######  Lemma 3.1.1.

The following statements are true:

  * i)

$[S_{4}({\rm adm}_{1,\,1})]^{S_{4}}=\langle[p_{1,\,1}]\rangle$ with
$p_{1,\,1}:=\sum_{1\leq j\leq 12}{\rm adm}_{1,\,j}.$

  * ii)

$[S_{4}({\rm adm}_{1,\,13})]^{S_{4}}=\langle[p_{1,\,2}]\rangle$ with
$p_{1,\,2}:=\sum_{13\leq j\leq 18}{\rm adm}_{1,\,j}.$

  * iii)

$[S_{4}({\rm adm}_{1,\,19},\ {\rm
adm}_{1,\,27})]^{S_{4}}=\langle[p_{1,\,3}]\rangle$ with
$p_{1,\,3}:=\sum_{\mathbb{J}\setminus\\{27,\ldots,31,34,35,36\\}}{\rm
adm}_{1,\,j},$
where the set $\mathbb{J}:=\\{19,20,\ldots,42\\}.$

###### Proof.

We prove Part iii) in detail. The others can be proved by the similar
computations.

It is easy to see that the set $\\{[{\rm adm}_{1,\,j}]:\,19\leq j\leq 42\\}$
is a basis of $S_{4}({\rm adm}_{1,\,19},\ {\rm adm}_{1,\,27}).$ Suppose that
$[f]\in[S_{4}({\rm adm}_{1,\,19},\ {\rm adm}_{1,\,27})]^{S_{4}},$ then
$f\equiv\sum_{19\leq j\leq 42}\gamma_{j}{\rm adm}_{1,\,j}$ in which
$\gamma_{j}\in\mathbb{Z}/2.$ Acting the homomorphisms $\theta_{i}$ for $1\leq
i\leq 3$ on both sides of this equality, we get

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\theta_{1}(f)&\equiv\gamma_{21}{\rm adm}_{1,\,19}+\gamma_{22}{\rm
adm}_{1,\,20}+\gamma_{19}{\rm adm}_{1,\,21}+\gamma_{20}{\rm
adm}_{1,\,22}+\gamma_{23}{\rm adm}_{1,\,23}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt&+\gamma_{24}{\rm adm}_{1,\,24}+\gamma_{28}{\rm
adm}_{1,\,27}+\gamma_{27}{\rm adm}_{1,\,28}+\gamma_{34}{\rm
adm}_{1,\,34}+\gamma_{37}{\rm adm}_{1,\,32}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt&+\gamma_{40}{\rm adm}_{1,\,33}+\gamma_{31}{\rm
adm}_{1,\,34}+\gamma_{38}{\rm adm}_{1,\,35}+\gamma_{39}{\rm
adm}_{1,\,36}+\gamma_{32}{\rm adm}_{1,\,37}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt&+\gamma_{35}{\rm adm}_{1,\,38}+\gamma_{36}{\rm
adm}_{1,\,39}+\gamma_{33}{\rm
adm}_{1,\,40}+\gamma_{25}t_{1}^{6}t_{2}t_{4}+\gamma_{26}t_{1}^{6}t_{2}t_{3}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&+\gamma_{29}t_{1}^{2}t_{2}t_{4}^{5}+\gamma_{30}t_{1}^{2}t_{2}t_{3}^{5}+\gamma_{41}t_{1}^{4}t_{2}^{3}t_{4}+\gamma_{42}t_{1}^{4}t_{2}^{3}t_{3},\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\theta_{2}(f)&\equiv\gamma_{19}{\rm
adm}_{1,\,19}+\gamma_{23}{\rm adm}_{1,\,21}+\gamma_{25}{\rm
adm}_{1,\,22}+\gamma_{21}{\rm adm}_{1,\,23}+\gamma_{26}{\rm adm}_{1,\,24}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt&+\gamma_{22}{\rm
adm}_{1,\,25}+\gamma_{24}{\rm adm}_{1,\,26}+\gamma_{29}{\rm
adm}_{1,\,28}+\gamma_{28}{\rm adm}_{1,\,29}+\gamma_{32}{\rm adm}_{1,\,31}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt&+\gamma_{31}{\rm
adm}_{1,\,32}+\gamma_{35}{\rm adm}_{1,\,34}+\gamma_{34}{\rm
adm}_{1,\,35}+\gamma_{38}{\rm adm}_{1,\,37}+\gamma_{37}{\rm adm}_{1,\,38}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt&+\gamma_{42}{\rm
adm}_{1,\,39}+\gamma_{41}{\rm adm}_{1,\,40}+\gamma_{40}{\rm
adm}_{1,\,41}+\gamma_{39}{\rm adm}_{1,\,42}+\gamma_{20}t_{2}^{6}t_{3}t_{4}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&+\gamma_{27}t_{2}^{2}t_{3}t_{4}^{5}+\gamma_{30}t_{1}t_{2}^{5}t_{3}^{2}+\gamma_{33}t_{2}^{4}t_{3}^{3}t_{4}+\gamma_{36}t_{1}t_{2}^{4}t_{3}^{3},\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\theta_{3}(f)&\equiv\gamma_{20}{\rm
adm}_{1,\,19}+\gamma_{19}{\rm adm}_{1,\,20}+\gamma_{22}{\rm
adm}_{1,\,21}+\gamma_{21}{\rm adm}_{1,\,22}+\gamma_{24}{\rm adm}_{1,\,23}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt&+\gamma_{23}{\rm
adm}_{1,\,24}+\gamma_{26}{\rm adm}_{1,\,25}+\gamma_{25}{\rm
adm}_{1,\,26}+\gamma_{30}{\rm adm}_{1,\,29}+\gamma_{29}{\rm adm}_{1,\,30}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt&+\gamma_{33}{\rm
adm}_{1,\,32}+\gamma_{32}{\rm adm}_{1,\,33}+\gamma_{36}{\rm
adm}_{1,\,35}+\gamma_{35}{\rm adm}_{1,\,36}+\gamma_{40}{\rm adm}_{1,\,37}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt&+\gamma_{39}{\rm
adm}_{1,\,38}+\gamma_{38}{\rm adm}_{1,\,39}+\gamma_{37}{\rm
adm}_{1,\,40}+\gamma_{42}{\rm adm}_{1,\,41}+\gamma_{41}{\rm adm}_{1,\,42}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&+\gamma_{27}t_{2}t_{3}^{5}t_{4}^{2}+\gamma_{28}t_{1}t_{3}^{5}t_{4}^{2}+\gamma_{31}t_{2}t_{3}^{4}t_{4}^{3}+\gamma_{34}t_{1}t_{3}^{4}t_{4}^{3}.\end{array}$

As well known, the (left) action of Steenrod algebra on $P^{\otimes
4}_{n_{s}}$ is determined by

$Sq^{n}(t_{j})=\left\\{\begin{array}[]{lll}t_{j}&\mbox{if}&n=0,\\\
t_{j}^{2}&\mbox{if}&n=1,\\\ 0&\mbox{if}&n>1,\end{array}\right.$

and Cartan’s formula $Sq^{n}(fg)=\sum_{k+\ell=n}Sq^{k}(f)Sq^{\ell}(g)$ for all
$f,\,g\in P^{\otimes 4}_{n_{s}}.$ Using this action, we have

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0ptt_{2}t_{3}^{4}t_{4}^{3}&=Sq^{2}(t_{2}t_{3}^{2}t_{4}^{3})+{\rm
adm}_{1,\,27}\mod(\overline{\mathcal{A}}P^{\otimes 4}_{n_{1}}),\\\ \vskip
6.0pt plus 2.0pt minus
2.0ptt_{2}t_{3}^{5}t_{4}^{2}&=Sq^{2}(t_{2}t_{3}^{3}t_{4}^{2})+{\rm
adm}_{1,\,31}\mod(\overline{\mathcal{A}}P^{\otimes 4}_{n_{1}}),\\\ \vskip
6.0pt plus 2.0pt minus
2.0ptt_{2}^{2}t_{3}t_{4}^{5}&=Sq^{1}(t_{2}t_{3}t_{4}^{5})+{\rm
adm}_{1,\,19}+{\rm adm}_{1,\,27}\mod(\overline{\mathcal{A}}P^{\otimes
4}_{n_{1}}),\\\ \vskip 6.0pt plus 2.0pt minus
2.0ptt_{2}^{4}t_{3}^{3}t_{4}&=Sq^{1}(t_{2}t_{3}^{5}t_{4})+Sq^{2}(t_{2}^{2}t_{3}^{3}t_{4}+t_{2}t_{3}^{3}t_{4}^{2})+{\rm
adm}_{1,\,20}+{\rm adm}_{1,\,31}\mod(\overline{\mathcal{A}}P^{\otimes
4}_{n_{1}}),\\\ \vskip 6.0pt plus 2.0pt minus
2.0ptt_{2}^{6}t_{3}t_{4}&=Sq^{1}(t_{2}^{5}t_{3}t_{4})+Sq^{2}(t_{2}^{3}t_{3}^{2}t_{4}+t_{2}^{3}t_{3}t_{4}^{2})+{\rm
adm}_{1,\,32}+{\rm adm}_{1,\,33}\mod(\overline{\mathcal{A}}P^{\otimes
4}_{n_{1}}),\\\ \vskip 6.0pt plus 2.0pt minus
2.0ptt_{1}t_{3}^{4}t_{4}^{3}&=Sq^{2}(t_{1}t_{3}^{2}t_{4}^{3})+{\rm
adm}_{1,\,28}\mod(\overline{\mathcal{A}}P^{\otimes 4}_{n_{1}}),\\\ \vskip
6.0pt plus 2.0pt minus
2.0ptt_{1}t_{3}^{5}t_{4}^{2}&=Sq^{2}(t_{1}t_{3}^{3}t_{4}^{2})+{\rm
adm}_{1,\,34}\mod(\overline{\mathcal{A}}P^{\otimes 4}_{n_{1}}),\\\ \vskip
6.0pt plus 2.0pt minus
2.0ptt_{1}t_{2}^{4}t_{3}^{3}&=Sq^{2}(t_{1}t_{2}^{2}t_{3}^{3})+{\rm
adm}_{1,\,30}\mod(\overline{\mathcal{A}}P^{\otimes 4}_{n_{1}}),\\\ \vskip
6.0pt plus 2.0pt minus
2.0ptt_{1}t_{2}^{5}t_{3}^{2}&=Sq^{2}(t_{1}t_{2}^{3}t_{3}^{2})+{\rm
adm}_{1,\,36}\mod(\overline{\mathcal{A}}P^{\otimes 4}_{n_{1}}),\\\ \vskip
6.0pt plus 2.0pt minus
2.0ptt_{1}^{2}t_{2}t_{4}^{5}&=Sq^{1}(t_{1}t_{2}t_{4}^{5})+{\rm
adm}_{1,\,23}+{\rm adm}_{1,\,29}\mod(\overline{\mathcal{A}}P^{\otimes
4}_{n_{1}}),\\\ \vskip 6.0pt plus 2.0pt minus
2.0ptt_{1}^{2}t_{2}t_{3}^{5}&=Sq^{1}(t_{1}t_{2}t_{3}^{5})+{\rm
adm}_{1,\,24}+{\rm adm}_{1,\,30}\mod(\overline{\mathcal{A}}P^{\otimes
4}_{n_{1}}),\\\ \vskip 6.0pt plus 2.0pt minus
2.0ptt_{1}^{4}t_{2}^{3}t_{4}&=Sq^{1}(t_{1}t_{2}^{5}t_{4})+Sq^{2}(t_{1}^{2}t_{2}^{3}t_{4}+t_{1}t_{2}^{3}t_{4}^{2})+{\rm
adm}_{1,\,25}+{\rm adm}_{1,\,35}\mod(\overline{\mathcal{A}}P^{\otimes
4}_{n_{1}}),\\\ \vskip 6.0pt plus 2.0pt minus
2.0ptt_{1}^{4}t_{2}^{3}t_{3}&=Sq^{1}(t_{1}t_{2}^{5}t_{3})+Sq^{2}(t_{1}^{2}t_{2}^{3}t_{3}+t_{1}t_{2}^{3}t_{3}^{2})+{\rm
adm}_{1,\,26}+{\rm adm}_{1,\,36}\mod(\overline{\mathcal{A}}P^{\otimes
4}_{n_{1}}),\\\ \vskip 6.0pt plus 2.0pt minus
2.0ptt_{1}^{6}t_{2}t_{4}&=Sq^{1}(t_{1}^{5}t_{2}t_{4})+Sq^{2}(t_{1}^{3}t_{2}^{2}t_{4}+t_{1}^{3}t_{2}t_{4}^{2})+{\rm
adm}_{1,\,38}+{\rm adm}_{1,\,41}\mod(\overline{\mathcal{A}}P^{\otimes
4}_{n_{1}}),\\\ \vskip 6.0pt plus 2.0pt minus
2.0ptt_{1}^{6}t_{2}t_{3}&=Sq^{1}(t_{1}^{5}t_{2}t_{3})+Sq^{2}(t_{1}^{3}t_{2}^{2}t_{3}+t_{1}^{3}t_{2}t_{3}^{2})+{\rm
adm}_{1,\,39}+{\rm adm}_{1,\,42}\mod(\overline{\mathcal{A}}P^{\otimes
4}_{n_{1}}).\end{array}$

Combining the above computations and the fact that $\theta_{i}(f)+f\equiv 0$
for $1\leq i\leq 3,$ we obtain $\gamma_{j}=0$ for
$j\in\mathbb{J}_{1}=\\{27,\ldots,31,34,35,36\\}$ and $\gamma_{j}=\gamma_{19}$
for $j\not\in\mathbb{J}_{1}.$ This finishes the proof of the lemma. ∎

######  Lemma 3.1.2.

There is a direct summand decomposition of the $S_{4}$-submodules:

$\widehat{{\rm Ker}_{n_{1}}}=S_{4}({\rm adm}_{1,\,43})\bigoplus S_{4}({\rm
adm}_{1,\,46}),$

where

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus 2.0ptS_{4}({\rm
adm}_{1,\,43})&=\langle[{\rm adm}_{1,\,43}],[{\rm adm}_{1,\,44}],[{\rm
adm}_{1,\,45}]\rangle,\ \mbox{with }\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt&\quad{\rm adm}_{1,\,43}=t_{1}t_{2}t_{3}^{2}t_{4}^{4},\,{\rm
adm}_{1,\,44}=t_{1}t_{2}^{2}t_{3}t_{4}^{4},\,{\rm
adm}_{1,\,45}=t_{1}t_{2}^{2}t_{3}^{4}t_{4},\\\ \vskip 6.0pt plus 2.0pt minus
2.0ptS_{4}({\rm adm}_{1,\,46})&=\langle[{\rm adm}_{1,\,46}],[{\rm
adm}_{1,\,47}],[{\rm adm}_{1,\,48}],[{\rm adm}_{1,\,49}]\rangle\
\mbox{with}\\\ &\quad{\rm
adm}_{1,\,46}=t_{1}t_{2}^{2}t_{3}^{2}t_{4}^{3},\,{\rm
adm}_{1,\,47}=t_{1}t_{2}^{2}t_{3}^{3}t_{4}^{2},\,{\rm
adm}_{1,\,48}=t_{1}t_{2}^{3}t_{3}^{2}t_{4}^{2},\,{\rm
adm}_{1,\,48}=t_{1}^{3}t_{2}t_{3}^{2}t_{4}^{2}.\end{array}$

Moreover, $[S_{4}({\rm adm}_{1,\,43})]^{S_{4}}=\langle[p_{1,\,4}]\rangle$ with
$p_{1,\,4}:=\sum_{43\leq j\leq 45}{\rm adm}_{1,\,j}$ and $[S_{4}({\rm
adm}_{1,\,46})]^{S_{4}}=0.$

###### Proof.

It is known, from a result of Sum [43, 44], that the set $\\{[{\rm
adm}_{1,\,j}]:\,43\leq j\leq 49\\}$ is an admissible monomial basis of
$\widehat{{\rm Ker}_{n_{1}}}.$ So, it is easy to see that $\widehat{{\rm
Ker}_{n_{1}}}\cong S_{4}({\rm adm}_{1,\,43})\bigoplus S_{4}({\rm
adm}_{1,\,46}).$

Now if $[g]\in[S_{4}({\rm adm}_{1,\,43})]^{S_{4}}$ and $[h]\in[S_{4}({\rm
adm}_{1,\,46})]^{S_{4}},$ then

$g\equiv\sum_{43\leq j\leq 45}\gamma_{j}{\rm adm}_{1,\,j},\ \
h\equiv\sum_{46\leq j\leq 49}\gamma_{j}{\rm adm}_{1,\,j},$

where $\gamma_{j}\in\mathbb{Z}/2.$ For each $1\leq i\leq 3,$ direct
calculating the equalities $\theta_{i}(g)$ and $\theta_{i}(h)$ in the terms
${\rm adm}_{1,\,j}$ for $43\leq j\leq 49$ mod
($\overline{\mathcal{A}}P^{\otimes 4}_{n_{1}}$), and using the relations
$\theta_{i}(g)+g\equiv 0$ and $\theta_{i}(h)+h\equiv 0,$ we get

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\theta_{1}(g)+g&\equiv\gamma_{43}({\rm adm}_{1,\,44}+{\rm
adm}_{1,\,45})\equiv 0,\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\theta_{2}(g)+g&\equiv(\gamma_{43}+\gamma_{44})({\rm adm}_{1,\,43}+{\rm
adm}_{1,\,44})\equiv 0,\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\theta_{3}(g)+g&\equiv(\gamma_{44}+\gamma_{45})({\rm adm}_{1,\,43}+{\rm
adm}_{1,\,44})\equiv 0,\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\theta_{1}(h)+h&\equiv(\gamma_{46}+\gamma_{47}){\rm
adm}_{1,\,43}+(\gamma_{48}+\gamma_{49})({\rm adm}_{1,\,48}+{\rm
adm}_{1,\,49})\equiv 0,\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\theta_{2}(h)+h&\equiv\gamma_{49}({\rm adm}_{1,\,43}+{\rm
adm}_{1,\,44})+(\gamma_{47}+\gamma_{48})({\rm adm}_{1,\,47}+{\rm
adm}_{1,\,48})\equiv 0,\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\theta_{3}(h)+h&\equiv(\gamma_{46}+\gamma_{47})({\rm adm}_{1,\,46}+{\rm
adm}_{1,\,47})\equiv 0.\end{array}$

From the equalities above, we get $\gamma_{43}=\gamma_{44}=\gamma_{45}$ and
$\gamma_{j}=0$ for $46\leq j\leq 49.$ The lemma follows. ∎

The case $s\geq 2$. By Sum [43, 44], $\underline{{\rm Ker}_{n_{s}}}$ has a
basis consisting of all the classes represented by the following admissible
monomials ${\rm adm}_{s,\,j}$:

For $s\geq 2,$

${\rm adm}_{s,\,1}=t_{2}t_{3}^{2^{s}-2}t_{4}^{2^{s+2}-1}$, |  ${\rm adm}_{s,\,2}=t_{2}t_{3}^{2^{s+2}-1}t_{4}^{2^{s}-2}$, |  ${\rm adm}_{s,\,3}=t_{2}^{2^{s+2}-1}t_{3}t_{4}^{2^{s}-2}$,
---|---|---
${\rm adm}_{s,\,4}=t_{1}t_{3}^{2^{s}-2}t_{4}^{2^{s+2}-1}$, |  ${\rm adm}_{s,\,5}=t_{1}t_{3}^{2^{s+2}-1}t_{4}^{2^{s}-2}$, |  ${\rm adm}_{s,\,6}=t_{1}^{2^{s+2}-1}t_{3}t_{4}^{2^{s}-2}$,
${\rm adm}_{s,\,7}=t_{1}t_{2}^{2^{s}-2}t_{4}^{2^{s+2}-1}$, |  ${\rm adm}_{s,\,8}=t_{1}t_{2}^{2^{s+2}-1}t_{4}^{2^{s}-2}$, |  ${\rm adm}_{s,\,9}=t_{1}^{2^{s+2}-1}t_{2}t_{4}^{2^{s}-2}$,
${\rm adm}_{s,\,10}=t_{1}t_{2}^{2^{s}-2}t_{3}^{2^{s+2}-1}$, |  ${\rm adm}_{s,\,11}=t_{1}t_{2}^{2^{s+2}-1}t_{3}^{2^{s}-2}$, |  ${\rm adm}_{s,\,12}=t_{1}^{2^{s+2}-1}t_{2}t_{3}^{2^{s}-2}$,
${\rm adm}_{s,\,13}=t_{2}t_{3}^{2^{s}-1}t_{4}^{2^{s+2}-2}$, |  ${\rm adm}_{s,\,14}=t_{2}t_{3}^{2^{s+2}-2}t_{4}^{2^{s}-1}$, |  ${\rm adm}_{s,\,15}=t_{2}^{2^{s}-1}t_{3}t_{4}^{2^{s+2}-2}$,
${\rm adm}_{s,\,16}=t_{1}t_{3}^{2^{s}-1}t_{4}^{2^{s+2}-2}$, |  ${\rm adm}_{s,\,17}=t_{1}t_{3}^{2^{s+2}-2}t_{4}^{2^{s}-1}$, |  ${\rm adm}_{s,\,18}=t_{1}^{2^{s}-1}t_{3}t_{4}^{2^{s+2}-2}$,
${\rm adm}_{s,\,19}=t_{1}t_{2}^{2^{s}-1}t_{4}^{2^{s+2}-2}$, |  ${\rm adm}_{s,\,20}=t_{1}t_{2}^{2^{s+2}-2}t_{4}^{2^{s}-1}$, |  ${\rm adm}_{s,\,21}=t_{1}^{2^{s}-1}t_{2}t_{4}^{2^{s+2}-2}$,
${\rm adm}_{s,\,22}=t_{1}t_{2}^{2^{s}-1}t_{3}^{2^{s+2}-2}$, |  ${\rm adm}_{s,\,23}=t_{1}t_{2}^{2^{s+2}-2}t_{3}^{2^{s}-1}$, |  ${\rm adm}_{s,\,24}=t_{1}^{2^{s}-1}t_{2}t_{3}^{2^{s+2}-2}$,
${\rm adm}_{s,\,25}=t_{3}^{2^{s}-1}t_{4}^{2^{s+2}-1}$, |  ${\rm adm}_{s,\,26}=t_{3}^{2^{s+2}-1}t_{4}^{2^{s}-1}$, |  ${\rm adm}_{s,\,27}=t_{2}^{2^{s}-1}t_{4}^{2^{s+2}-1}$,
${\rm adm}_{s,\,28}=t_{2}^{2^{s}-1}t_{3}^{2^{s+2}-1}$, |  ${\rm adm}_{s,\,29}=t_{2}^{2^{s+2}-1}t_{4}^{2^{s}-1}$, |  ${\rm adm}_{s,\,30}=t_{2}^{2^{s+2}-1}t_{3}^{2^{s}-1}$,
${\rm adm}_{s,\,31}=t_{1}^{2^{s}-1}t_{4}^{2^{s+2}-1}$, |  ${\rm adm}_{s,\,32}=t_{1}^{2^{s}-1}t_{3}^{2^{s+2}-1}$, |  ${\rm adm}_{s,\,33}=t_{1}^{2^{s+2}-1}t_{4}^{2^{s}-1}$,
${\rm adm}_{s,\,34}=t_{1}^{2^{s+2}-1}t_{3}^{2^{s}-1}$, |  ${\rm adm}_{s,\,35}=t_{1}^{2^{s}-1}t_{2}^{2^{s+2}-1}$, |  ${\rm adm}_{s,\,36}=t_{1}^{2^{s+2}-1}t_{2}^{2^{s}-1}$,
${\rm adm}_{s,\,37}=t_{2}t_{3}^{2^{s+1}-2}t_{4}^{2^{s+2}-2^{s}-1}$, |  ${\rm adm}_{s,\,38}=t_{1}t_{3}^{2^{s+1}-2}t_{4}^{2^{s+2}-2^{s}-1}$, |  ${\rm adm}_{s,\,39}=t_{1}t_{2}^{2^{s+1}-2}t_{4}^{2^{s+2}-2^{s}-1}$,
${\rm adm}_{s,\,40}=t_{1}t_{2}^{2^{s+1}-2}t_{3}^{2^{s+2}-2^{s}-1}$, |  ${\rm adm}_{s,\,41}=t_{2}t_{3}^{2^{s+1}-1}t_{4}^{2^{s+2}-2^{s}-2}$, |  ${\rm adm}_{s,\,42}=t_{2}^{2^{s+1}-1}t_{3}t_{4}^{2^{s+2}-2^{s}-2}$,
${\rm adm}_{s,\,43}=t_{1}t_{3}^{2^{s+1}-1}t_{4}^{2^{s+2}-2^{s}-2}$, |  ${\rm adm}_{s,\,44}=t_{1}^{2^{s+1}-1}t_{3}t_{4}^{2^{s+2}-2^{s}-2}$, |  ${\rm adm}_{s,\,45}=t_{1}t_{2}^{2^{s+1}-1}t_{4}^{2^{s+2}-2^{s}-2}$,
${\rm adm}_{s,\,46}=t_{1}^{2^{s+1}-1}t_{2}t_{4}^{2^{s+2}-2^{s}-2}$, |  ${\rm adm}_{s,\,47}=t_{1}t_{2}^{2^{s+1}-1}t_{3}^{2^{s+2}-2^{s}-2}$, |  ${\rm adm}_{s,\,48}=t_{1}^{2^{s+1}-1}t_{2}t_{3}^{2^{s+2}-2^{s}-2}$,
${\rm adm}_{s,\,49}=t_{3}^{2^{s+1}-1}t_{4}^{2^{s+2}-2^{s}-1}$, |  ${\rm adm}_{s,\,50}=t_{2}^{2^{s+1}-1}t_{4}^{2^{s+2}-2^{s}-1}$, |  ${\rm adm}_{s,\,51}=t_{2}^{2^{s+1}-1}t_{3}^{2^{s+2}-2^{s}-1}$,
${\rm adm}_{s,\,52}=t_{1}^{2^{s+1}-1}t_{4}^{2^{s+2}-2^{s}-1}$, |  ${\rm adm}_{s,\,53}=t_{1}^{2^{s+1}-1}t_{3}^{2^{s+2}-2^{s}-1}$, |  ${\rm adm}_{s,\,54}=t_{1}^{2^{s+1}-1}t_{2}^{2^{s+2}-2^{s}-1}$,
${\rm adm}_{s,\,55}=t_{2}^{3}t_{3}^{2^{s+2}-3}t_{4}^{2^{s}-2}$, |  ${\rm adm}_{s,\,56}=t_{1}^{3}t_{3}^{2^{s+2}-3}t_{4}^{2^{s}-2}$, |  ${\rm adm}_{s,\,57}=t_{1}^{3}t_{2}^{2^{s+2}-3}t_{4}^{2^{s}-2}$,
${\rm adm}_{s,\,58}=t_{1}^{3}t_{2}^{2^{s+2}-3}t_{3}^{2^{s}-2}$, |  ${\rm adm}_{s,\,59}=t_{2}^{3}t_{3}^{2^{s+1}-3}t_{4}^{2^{s+2}-2^{s}-2}$, |  ${\rm adm}_{s,\,60}=t_{1}^{3}t_{3}^{2^{s+1}-3}t_{4}^{2^{s+2}-2^{s}-2}$,
${\rm adm}_{s,\,61}=t_{1}^{3}t_{2}^{2^{s+1}-3}t_{4}^{2^{s+2}-2^{s}-2}$, |  ${\rm adm}_{s,\,62}=t_{1}^{3}t_{2}^{2^{s+1}-3}t_{3}^{2^{s+2}-2^{s}-2}$. |

For $s=2,$

${\rm adm}_{2,\,63}=t_{2}^{3}t_{3}^{3}t_{4}^{12}$, |  ${\rm adm}_{2,\,64}=t_{1}^{3}t_{3}^{3}t_{4}^{12}$, |  ${\rm adm}_{2,\,65}=t_{1}^{3}t_{2}^{3}t_{4}^{12}$, |  ${\rm adm}_{2,\,66}=t_{1}^{3}t_{2}^{3}t_{3}^{12}$.
---|---|---|---

For $s\geq 3,$

${\rm adm}_{s,\,63}=t_{2}^{3}t_{3}^{2^{s}-3}t_{4}^{2^{s+2}-2}$, |  ${\rm adm}_{s,\,64}=t_{1}^{3}t_{3}^{2^{s}-3}t_{4}^{2^{s+2}-2}$, |  ${\rm adm}_{s,\,65}=t_{1}^{3}t_{2}^{2^{s}-3}t_{4}^{2^{s+2}-2}$,
---|---|---
${\rm adm}_{s,\,66}=t_{1}^{3}t_{2}^{2^{s}-3}t_{3}^{2^{s+2}-2}$. |  |

######  Lemma 3.1.3.

For each $s\geq 2,$ we have $\dim[\underline{{\rm Ker}_{n_{s}}}]^{S_{4}}\leq
6.$

###### Proof.

By direct computations using the admissible basis of $\underline{{\rm
Ker}_{n_{s}}}$ above, we notice that

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus 2.0pt&S_{4}({\rm
adm}_{s,\,1})=\langle\\{[{\rm adm}_{s,\,j}]:\,1\leq j\leq 12\\}\rangle,\\\
&S_{4}({\rm adm}_{s,\,13},\ {\rm adm}_{s,\,37},\ {\rm adm}_{s,\,59},\ {\rm
adm}_{s,\,63})=\langle\\{[{\rm adm}_{s,\,j}]:\,j\in\mathbb{J}\\}\rangle,\
\mbox{where}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt&\quad\mathbb{J}=\\{13,\ldots,24,37,\ldots,48,55,\ldots,66\\},\\\ \vskip
6.0pt plus 2.0pt minus 2.0pt&S_{4}({\rm adm}_{s,\,25})=\langle\\{[{\rm
adm}_{s,\,j}]:\,25\leq j\leq 36\\}\rangle,\ \ S_{4}({\rm
adm}_{s,\,49})=\langle\\{[{\rm adm}_{s,\,j}]:\,49\leq j\leq 54\\}\rangle,\\\
\end{array}$

are $S_{4}$-submodules of $\underline{{\rm Ker}_{n_{s}}}.$ Hence, we have an
isomorphism

$\underline{{\rm Ker}_{n_{s}}}\cong S_{4}({\rm adm}_{s,\,1})\bigoplus
S_{4}({\rm adm}_{s,\,13},{\rm adm}_{s,\,37},{\rm adm}_{s,\,59},{\rm
adm}_{s,\,63})\bigoplus S_{4}({\rm adm}_{s,\,25})\bigoplus S_{4}({\rm
adm}_{s,\,49}).$

Then, by the similar techniques as in the proof of Lemma 3.1.1, it may be
concluded that

  * i)

$[S_{4}({\rm adm}_{s,\,1})]^{S_{4}}=\langle[p_{s,\,1}]\rangle$ with
$p_{s,\,1}:=\sum_{1\leq j\leq 12}{\rm adm}_{s,\,j}$

  * ii)

$\begin{array}[]{ll}&[S_{4}({\rm adm}_{s,\,13},{\rm adm}_{s,\,37},{\rm
adm}_{s,\,59},{\rm adm}_{s,\,63})\bigoplus S_{4}({\rm
adm}_{s,\,25})]^{S_{4}}\\\ &=\left\\{\begin{array}[]{ll}\vskip 6.0pt plus
2.0pt minus
2.0pt\langle[p_{2,\,2}+p_{2,\,3}+p_{2,\,4}+p_{2,\,5}],[p_{2,\,3}+p_{2,\,4}+p_{2,\,6}],[p_{2,\,2}+p_{2,\,7}]\rangle&\mbox{if
$s=2$},\\\
\langle[p_{s,\,2}+p_{s,\,3}],[p_{s,\,2}+p_{s,\,4}],[p_{s,\,2}+p_{s,\,5}]\rangle&\mbox{if
$s\geq 3$},\end{array}\right.\end{array}$
where

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus 2.0ptp_{2,\,2}&={\rm
adm}_{2,\,13}+{\rm adm}_{2,\,16}+{\rm adm}_{2,\,18}+{\rm adm}_{2,\,19},\\\
\vskip 6.0pt plus 2.0pt minus 2.0ptp_{2,\,3}&={\rm adm}_{2,\,14}+{\rm
adm}_{2,\,17}+{\rm adm}_{2,\,20}+{\rm adm}_{2,\,21},\\\ \vskip 6.0pt plus
2.0pt minus 2.0ptp_{2,\,4}&={\rm adm}_{2,\,15}+{\rm adm}_{2,\,22}+{\rm
adm}_{2,\,23}+{\rm adm}_{2,\,24},\\\ \vskip 6.0pt plus 2.0pt minus
2.0ptp_{2,\,5}&=\sum_{37\leq j\leq 48}{\rm adm}_{2,\,j},\ \
p_{2,\,6}=\sum_{55\leq j\leq 58}{\rm adm}_{2,\,j}+\sum_{63\leq j\leq 66}{\rm
adm}_{2,\,j},\\\ \vskip 6.0pt plus 2.0pt minus 2.0ptp_{2,\,7}&=\sum_{59\leq
j\leq 62}{\rm adm}_{2,\,j},\ \ p_{s,\,2}=\sum_{13\leq j\leq 24}{\rm
adm}_{s,\,j},\ \ p_{s,\,3}=\sum_{37\leq j\leq 48}{\rm adm}_{s,\,j},\\\ \vskip
6.0pt plus 2.0pt minus 2.0ptp_{s,\,4}&=\sum_{55\leq j\leq 58}{\rm
adm}_{s,\,j},\ \ p_{s,\,5}=\sum_{59\leq j\leq 66}{\rm
adm}_{s,\,j}.\end{array}$

  * iii)

$[S_{4}({\rm adm}_{s,\,25})]^{S_{4}}=\langle[p_{s,\,6}]\rangle$ with
$p_{s,\,6}:=\sum_{25\leq j\leq 36}{\rm adm}_{s,\,j}.$

  * iv)

$[S_{4}({\rm adm}_{s,\,49})]^{S_{4}}=\langle[p_{s,\,7}]\rangle$ with
$p_{s,\,7}:=\sum_{49\leq j\leq 54}{\rm adm}_{s,\,j}.$

These imply that

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus 2.0pt&\dim[S_{4}({\rm
adm}_{s,\,j})]^{S_{4}}=1\ \mbox{for}\ j=1,\,25,\,49,\\\ &\dim[S_{4}({\rm
adm}_{s,\,13},{\rm adm}_{s,\,37},{\rm adm}_{s,\,59},{\rm
adm}_{s,\,63})]^{S_{4}}=3.\end{array}$

Now, the lemma follows from the fact that

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus 2.0pt\dim[\underline{{\rm
Ker}_{n_{s}}}]^{S_{4}}&\leq\dim[S_{4}({\rm
adm}_{s,\,1})]^{S_{4}}+\dim[S_{4}({\rm adm}_{s,\,13},{\rm adm}_{s,\,37},{\rm
adm}_{s,\,59},{\rm adm}_{s,\,63})]^{S_{4}}\\\ &\quad+\dim[S_{4}({\rm
adm}_{s,\,25})]^{S_{4}}+\dim\dim[S_{4}({\rm
adm}_{s,\,49})]^{S_{4}}.\end{array}$

∎

From a result in Sum [43, 44], $\widehat{{\rm Ker}_{n_{s}}}$ has a basis
consisting of all the classes represented by the following admissible
monomials ${\rm adm}_{s,\,j}$:

For $s\geq 2,$

${\rm adm}_{s,\,67}=t_{1}t_{2}t_{3}^{2^{s}-2}t_{4}^{2^{s+2}-2}$, |  ${\rm adm}_{s,\,68}=t_{1}t_{2}t_{3}^{2^{s+2}-2}t_{4}^{2^{s}-2}$, |  ${\rm adm}_{s,\,69}=t_{1}t_{2}^{2^{s}-2}t_{3}t_{4}^{2^{s+2}-2}$,
---|---|---
${\rm adm}_{s,\,70}=t_{1}t_{2}^{2^{s+2}-2}t_{3}t_{4}^{2^{s}-2}$, |  ${\rm adm}_{s,\,71}=t_{1}t_{2}^{2}t_{3}^{2^{s+2}-3}t_{4}^{2^{s}-2}$, |  ${\rm adm}_{s,\,72}=t_{1}t_{2}t_{3}^{2^{s+1}-2}t_{4}^{2^{s+2}-2^{s}-2}$,
${\rm adm}_{s,\,73}=t_{1}t_{2}^{2^{s+1}-2}t_{3}t_{4}^{2^{s+2}-2^{s}-2}$, |  ${\rm adm}_{s,\,74}=t_{1}t_{2}^{2}t_{3}^{2^{s}-1}t_{4}^{2^{s+2}-4}$, |  ${\rm adm}_{s,\,75}=t_{1}t_{2}^{2}t_{3}^{2^{s+2}-4}t_{4}^{2^{s}-1}$,
${\rm adm}_{s,\,76}=t_{1}t_{2}^{3}t_{3}^{2^{s}-2}t_{4}^{2^{s+2}-4}$, |  ${\rm adm}_{s,\,77}=t_{1}t_{2}^{3}t_{3}^{2^{s+2}-4}t_{4}^{2^{s}-2}$, |  ${\rm adm}_{s,\,78}=t_{1}^{3}t_{2}t_{3}^{2^{s}-2}t_{4}^{2^{s+2}-4}$,
${\rm adm}_{s,\,79}=t_{1}^{3}t_{2}t_{3}^{2^{s+2}-4}t_{4}^{2^{s}-2}$, |  ${\rm adm}_{s,\,80}=t_{1}t_{2}^{2}t_{3}^{2^{s+1}-4}t_{4}^{2^{s+2}-2^{s}-1}$, |  ${\rm adm}_{s,\,81}=t_{1}t_{2}^{2}t_{3}^{2^{s+1}-3}t_{4}^{2^{s+2}-2^{s}-2}$,
${\rm adm}_{s,\,82}=t_{1}t_{2}^{2}t_{3}^{2^{s+1}-1}t_{4}^{2^{s+2}-2^{s}-2}$, |  ${\rm adm}_{s,\,83}=t_{1}t_{2}^{2^{s+1}-1}t_{3}^{2}t_{4}^{2^{s+2}-2^{s}-2}$, |  ${\rm adm}_{s,\,84}=t_{1}^{2^{s+1}-1}t_{2}t_{3}^{2}t_{4}^{2^{s+2}-2^{s}-2}$,
${\rm adm}_{s,\,85}=t_{1}t_{2}^{3}t_{3}^{2^{s+1}-4}t_{4}^{2^{s+2}-2^{s}-2}$, |  ${\rm adm}_{s,\,86}=t_{1}^{3}t_{2}t_{3}^{2^{s+1}-4}t_{4}^{2^{s+2}-2^{s}-2}$, |  ${\rm adm}_{s,\,87}=t_{1}t_{2}^{3}t_{3}^{2^{s+1}-2}t_{4}^{2^{s+2}-2^{s}-2}$,
${\rm adm}_{s,\,88}=t_{1}^{3}t_{2}t_{3}^{2^{s+1}-2}t_{4}^{2^{s+2}-2^{s}-2}$, |  ${\rm adm}_{s,\,89}=t_{1}^{3}t_{2}^{2^{s+1}-3}t_{3}^{2}t_{4}^{2^{s+2}-2^{s}-2}$. |

For $s=2,$ ${\rm adm}_{2,\,90}=t_{1}^{3}t_{2}^{3}t_{3}^{4}t_{4}^{8}$, ${\rm
adm}_{2,\,91}=t_{1}^{3}t_{2}^{5}t_{3}^{8}t_{4}^{2}$.

For $s\geq 3,$

${\rm adm}_{s,\,90}=t_{1}t_{2}^{2}t_{3}^{2^{s}-4}t_{4}^{2^{s+2}-1}$, |  ${\rm adm}_{s,\,91}=t_{1}t_{2}^{2}t_{3}^{2^{s+2}-1}t_{4}^{2^{s}-4}$, |  ${\rm adm}_{s,\,92}=t_{1}t_{2}^{2^{s+2}-1}t_{3}^{2}t_{4}^{2^{s}-4}$,
---|---|---
${\rm adm}_{s,\,93}=t_{1}^{2^{s+2}-1}t_{2}t_{3}^{2}t_{4}^{2^{s}-4}$, |  ${\rm adm}_{s,\,94}=t_{1}t_{2}^{2}t_{3}^{2^{s}-3}t_{4}^{2^{s+2}-2}$, |  ${\rm adm}_{s,\,95}=t_{1}t_{2}^{3}t_{3}^{2^{s}-4}t_{4}^{2^{s+2}-2}$,
${\rm adm}_{s,\,96}=t_{1}t_{2}^{3}t_{3}^{2^{s+2}-2}t_{4}^{2^{s}-4}$, |  ${\rm adm}_{s,\,97}=t_{1}^{3}t_{2}t_{3}^{2^{s}-4}t_{4}^{2^{s+2}-2}$, |  ${\rm adm}_{s,\,98}=t_{1}^{3}t_{2}t_{3}^{2^{s+2}-2}t_{4}^{2^{s}-4}$,
${\rm adm}_{s,\,99}=t_{1}t_{2}^{2^{s}-1}t_{3}^{2}t_{4}^{2^{s+2}-4}$, |  ${\rm adm}_{s,\,100}=t_{1}^{2^{s}-1}t_{2}t_{3}^{2}t_{4}^{2^{s+2}-4}$, |  ${\rm adm}_{s,\,101}=t_{1}^{3}t_{2}^{2^{s+2}-3}t_{3}^{2}t_{4}^{2^{s}-4}$,
${\rm adm}_{s,\,102}=t_{1}^{3}t_{2}^{2^{s}-3}t_{3}^{2}t_{4}^{2^{s+2}-4}$, |  ${\rm adm}_{s,\,103}=t_{1}^{3}t_{2}^{5}t_{3}^{2^{s+2}-6}t_{4}^{2^{s}-4}$, |  ${\rm adm}_{s,\,104}=t_{1}^{3}t_{2}^{5}t_{3}^{2^{s+1}-6}t_{4}^{2^{s+2}-2^{s}-2}$.

For $s=3,$ ${\rm adm}_{3,\,105}=t_{1}^{3}t_{2}^{5}t_{3}^{6}t_{4}^{24}.$

For $s\geq 4,$ ${\rm
adm}_{s,\,105}=t_{1}^{3}t_{2}^{5}t_{3}^{2^{s}-6}t_{4}^{2^{s+2}-4}.$

######  Proposition 3.1.4.

Let $s$ be a positive integer. Then,

$[{\rm Ker}_{n_{s}}]^{GL_{4}}=\left\\{\begin{array}[]{ll}0&\mbox{if $s=1$},\\\
\langle[\widehat{p_{2,\,1}}]\rangle&\mbox{if $s=2$},\\\ \langle[\sum_{1\leq
j\leq 105}{\rm adm}_{s,\,j}]\rangle&\mbox{if $s\geq 3$},\\\
\end{array}\right.$

where

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\widehat{p_{2,\,1}}&={\rm adm}_{2,\,67}+{\rm adm}_{2,\,68}+{\rm
adm}_{2,\,72}+{\rm adm}_{2,\,76}+{\rm adm}_{2,\,77}\\\ &\quad+{\rm
adm}_{2,\,78}+{\rm adm}_{2,\,79}+{\rm adm}_{2,\,89}+{\rm
adm}_{2,\,91}.\end{array}$

###### Proof.

Firstly, we consider the case $s=1.$ Suppose that $[\rho]\in[{\rm
Ker}_{n_{1}}]^{GL_{4}}.$ Then , since $S_{4}\subset GL_{4},$ by Lemmas 3.1.1
and 3.1.2, we have
$\rho\equiv\beta_{1}p_{1,\,1}+\beta_{2}p_{1,\,2}+\beta_{3}p_{1,\,3}+\beta_{4}p_{1,\,4},$
where $\beta_{i}\in\mathbb{Z}/2.$ Direct calculating $\theta_{4}(\rho)$ in the
admissible terms ${\rm adm}_{1,\,j}$ for $1\leq j\leq 49$ and using the
relation $\theta_{4}(\rho)+\rho\equiv 0,$ we obtain
$\theta_{4}(\rho)+\rho\equiv[\beta_{1}{\rm
adm}_{1,\,3}+(\beta_{1}+\beta_{2}){\rm adm}_{1,\,9}+\beta_{3}{\rm
adm}_{1,\,19}+(\beta_{3}+\beta_{4}){\rm adm}_{1,\,32}+\ \mbox{other
terms}]\equiv 0.$ This equality implies
$\beta_{1}=\beta_{2}=\beta_{3}=\beta_{4}=0.$ Thus, $[{\rm
Ker}_{n_{1}}]^{GL_{4}}$ is trivial.

Next, for $s=2,$ we see that the set $\\{[{\rm adm}_{2,\,j}]:\,67\leq j\leq
91\\}$ is a basis of $\widehat{{\rm Ker}_{n_{2}}}.$ Assume that
$[f]\in[\widehat{{\rm Ker}_{n_{2}}}]^{S_{4}}.$ Then, $f\equiv\sum_{67\leq
j\leq 91}\gamma_{j}{\rm adm}_{2,\,j}$ with $\gamma_{j}\in\mathbb{Z}/2.$ Using
the relations $\theta_{i}(f)+f\equiv 0$ for $1\leq i\leq 3,$ it follows that

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\theta_{1}(f)+f&\equiv[(\gamma_{69}+\gamma_{73}){\rm
adm}_{2,\,67}+\gamma_{71}{\rm adm}_{2,\,68}+\gamma_{81}{\rm
adm}_{2,\,72}+(\gamma_{73}+\gamma_{76}+\gamma_{78}){\rm adm}_{2,\,76}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+(\gamma_{70}+\gamma_{77}+\gamma_{79}){\rm
adm}_{2,\,77}+(\gamma_{83}+\gamma_{84}){\rm
adm}_{2,\,83}+(\gamma_{85}+\gamma_{86}){\rm adm}_{2,\,85}\\\ \vskip 6.0pt plus
2.0pt minus 2.0pt&\quad+(\gamma_{87}+\gamma_{88}){\rm
adm}_{2,\,87}+\gamma_{70}{\rm
adm}_{2,\,89}+(\gamma_{70}+\gamma_{73}+\gamma_{89}+\gamma_{91}){\rm
adm}_{2,\,90}\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt&\quad+\mbox{other
terms}]\ \equiv 0,\\\ \end{array}$

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\theta_{2}(f)+f&\equiv[(\gamma_{67}+\gamma_{69}+\gamma_{75}+\gamma_{80}+\gamma_{86}+\gamma_{87}+\gamma_{90}+\gamma_{91}){\rm
adm}_{2,\,67}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+(\gamma_{67}+\gamma_{69}+\gamma_{75}+\gamma_{80}++\gamma_{87}+\gamma_{90}+\gamma_{91}){\rm
adm}_{2,\,69}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+(\gamma_{68}+\gamma_{70}+\gamma_{91}){\rm
adm}_{2,\,68}+(\gamma_{68}+\gamma_{70}+\gamma_{79}){\rm adm}_{2,\,70}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+(\gamma_{71}+\gamma_{77}+\gamma_{91}){\rm
adm}_{2,\,71}+(\gamma_{72}+\gamma_{73}+\gamma_{87}+\gamma_{90}+\gamma_{91}){\rm
adm}_{2,\,72}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+(\gamma_{72}+\gamma_{73}+\gamma_{86}+\gamma_{87}+\gamma_{90}+\gamma_{91}){\rm
adm}_{2,\,73}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+(\gamma_{74}+\gamma_{76}+\gamma_{77}+\gamma_{85}+\gamma_{87}+\gamma_{90}){\rm
adm}_{2,\,74}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+(\gamma_{71}+\gamma_{74}+\gamma_{76}+\gamma_{81}+\gamma_{86}+\gamma_{87}+\gamma_{90}+\gamma_{91}){\rm
adm}_{2,\,76}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+(\gamma_{71}+\gamma_{77}+\gamma_{79}){\rm
adm}_{2,\,77}+\gamma_{86}{\rm adm}_{2,\,78}+(\gamma_{79}+\gamma_{91}){\rm
adm}_{2,\,79}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+(\gamma_{81}+\gamma_{85}){\rm
adm}_{2,\,81}+(\gamma_{82}+\gamma_{83}){\rm
adm}_{2,\,82}+(\gamma_{88}+\gamma_{89}+\gamma_{91}){\rm adm}_{2,\,88}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+(\gamma_{79}+\gamma_{88}+\gamma_{89}){\rm
adm}_{2,\,89}+(\gamma_{79}+\gamma_{91}){\rm adm}_{2,\,91}\\\ \vskip 6.0pt plus
2.0pt minus 2.0pt&\quad+(\gamma_{79}+\gamma_{86}+\gamma_{91}){\rm
adm}_{2,\,90}+\ \mbox{other terms}]\ \equiv 0,\\\ \vskip 6.0pt plus 2.0pt
minus
2.0pt\theta_{3}(f)+f&\equiv[(\gamma_{67}+\gamma_{68}+\gamma_{71}+\gamma_{73}){\rm
adm}_{2,\,67}+(\gamma_{67}+\gamma_{68}+\gamma_{69}){\rm adm}_{2,\,68}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt&\quad(\gamma_{69}+\gamma_{71}+\gamma_{73}){\rm
adm}_{2,\,69}+(\gamma_{73}+\gamma_{81}){\rm
adm}_{2,\,72}+(\gamma_{74}+\gamma_{75}){\rm adm}_{2,\,74}\\\ \vskip 6.0pt plus
2.0pt minus
2.0pt&\quad+(\gamma_{70}+\gamma_{73}+\gamma_{76}+\gamma_{77}+\gamma_{83}){\rm
adm}_{2,\,76}+(\gamma_{78}+\gamma_{79}+\gamma_{84}){\rm adm}_{2,\,78}\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt&\quad+(\gamma_{80}+\gamma_{82}){\rm
adm}_{2,\,80}+(\gamma_{85}+\gamma_{87}){\rm
adm}_{2,\,85}+(\gamma_{86}+\gamma_{88}){\rm adm}_{2,\,86}\\\ \vskip 6.0pt plus
2.0pt minus 2.0pt&\quad+(\gamma_{89}+\gamma_{91}){\rm adm}_{2,\,89}\
+\mbox{other terms}]\ \equiv 0.\end{array}$

From these computations, it may be concluded that $[\widehat{{\rm
Ker}_{n_{2}}}]^{S_{4}}=\langle[\widehat{p_{2,\,1}}],[\widehat{p_{2,\,2}}],[\widehat{p_{2,\,3}}]\rangle,$
where

$\begin{array}[]{ll}\widehat{p_{2,\,1}}&:=\sum_{\gamma_{j}\in\mathbb{J}_{1}}{\rm
adm}_{2,\,j},\ \mbox{with\
$\mathbb{J}_{1}=\\{67,68,72,76,77,78,79,89,91\\},$}\\\\[2.84526pt]
\widehat{p_{2,\,2}}&:=\sum_{\gamma_{j}\in\mathbb{J}_{2}}{\rm adm}_{2,\,j},\
\mbox{with\ $\mathbb{J}_{2}=\\{74,75,76,78,80,82,83,84\\}$},\\\\[2.84526pt]
\widehat{p_{2,\,3}}&:=\sum_{\gamma_{j}\in\mathbb{J}_{3}}{\rm adm}_{2,\,j},\
\mbox{with\ $\mathbb{J}_{3}=\\{72,74,75,90\\}$}.\end{array}$

Now, suppose that $[\rho]\in[{\rm Ker}_{n_{2}}]^{GL_{4}},$ then combining the
proof of Lemma 3.1.3 and the fact that $S_{4}\subset GL_{4},$ we have

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\rho&\equiv\alpha_{1}p_{2,\,1}+\alpha_{2}\sum_{2\leq i\leq
5}p_{2,\,i}+\alpha_{3}(p_{2,\,3}+p_{2,\,4}+p_{2,\,6})\\\
&\quad+\alpha_{4}(p_{2,\,2}+p_{2,\,7})+\alpha_{5}p_{2,\,6}+\alpha_{6}p_{2,\,7}+\alpha_{7}\widehat{p_{2,\,1}}+\alpha_{8}\widehat{p_{2,\,2}}+\alpha_{9}\widehat{p_{2,\,3}}.\end{array}$

Direct calculating $\theta_{4}(\rho)$ in the admissible terms ${\rm
adm}_{2,\,j}$ for $1\leq j\leq 91$ and using the relation
$\theta_{4}(\rho)+\rho\equiv 0,$ we find that

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\theta_{4}(\rho)+\rho&\equiv[\alpha_{1}{\rm
adm}_{2,\,1}+(\gamma_{1}+\gamma_{6}){\rm
adm}_{2,\,8}+(\gamma_{1}+\gamma_{2}){\rm
adm}_{2,\,27}+(\gamma_{2}+\gamma_{4}+\gamma_{6}){\rm adm}_{2,\,29}\\\ \vskip
6.0pt plus 2.0pt minus 2.0pt&\quad+(\gamma_{2}+\gamma_{3}){\rm
adm}_{2,\,33}+(\gamma_{4}+\gamma_{8}){\rm adm}_{2,\,8}+\gamma_{5}{\rm
adm}_{2,\,59}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt&\quad+(\gamma_{4}+\gamma_{6}+\gamma_{8}+\gamma_{9}){\rm adm}_{2,\,14}+\
\mbox{other terms}]\ \equiv 0.\end{array}$

This equality shows that $\alpha_{i}=0$ for $i\neq 7.$ Therefore, $[{\rm
Ker}_{n_{2}}]^{GL_{4}}=\langle[\widehat{p_{2,\,1}}]\rangle.$

Finally, by using Lemma 3.1.3 and the similar techniques as above, for each
$s\geq 3,$ we get $[{\rm Ker}_{n_{s}}]^{GL_{4}}=\langle[\sum_{1\leq j\leq
105}{\rm adm}_{s,\,j}]\rangle.$ The proposition is proved. ∎

###### Proof of Theorem 4.2.1.

The theorem follows from the inequality (5), Proposition 3.1.4 and the fact
that $\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{\frac{n_{1}-4}{2}}^{\otimes 4}]^{*}$ is
trivial (see Sum [46]). ∎

###### Proof of Theorem 2.1.4.

It is known, from a theorem of Sum [46] that $\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{\frac{n_{2}-4}{2}}^{\otimes 4}]^{*}$ is
1-dimensional and generated by
$[x_{1}^{(0)}x_{2}^{(0)}x_{3}^{(0)}x_{4}^{(7)}].$ Combining this with the
ineequality (5) and Proposition 3.1.4, we deduce that
$\dim\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n_{2}}^{\otimes 4}]^{*}\leq 2.$ On the other
hand, consider the element

$\begin{array}[]{ll}\zeta&=(x_{1}^{(3)}x_{2}^{(5)}x_{3}^{(1)}x_{4}^{(9)}+x_{1}^{(3)}x_{2}^{(5)}x_{3}^{(2)}x_{4}^{(8)}+x_{1}^{(3)}x_{2}^{(6)}x_{3}^{(1)}x_{4}^{(8)}+x_{1}^{(3)}x_{2}^{(6)}x_{3}^{(2)}x_{4}^{(7)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(3)}x_{2}^{(5)}x_{3}^{(4)}x_{4}^{(6)}\\\
&\quad+x_{1}^{(3)}x_{2}^{(6)}x_{3}^{(3)}x_{4}^{(6)}+x_{1}^{(5)}x_{2}^{(6)}x_{3}^{(1)}x_{4}^{(6)}+x_{1}^{(3)}x_{2}^{(5)}x_{3}^{(5)}x_{4}^{(5)}+x_{1}^{(3)}x_{2}^{(6)}x_{3}^{(4)}x_{4}^{(5)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(5)}x_{2}^{(6)}x_{3}^{(2)}x_{4}^{(5)}\\\
&\quad+x_{1}^{(3)}x_{2}^{(9)}x_{3}^{(1)}x_{4}^{(5)}+x_{1}^{(5)}x_{2}^{(7)}x_{3}^{(1)}x_{4}^{(5)}+x_{1}^{(3)}x_{2}^{(9)}x_{3}^{(2)}x_{4}^{(4)}+x_{1}^{(5)}x_{2}^{(7)}x_{3}^{(2)}x_{4}^{(4)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(3)}x_{2}^{(10)}x_{3}^{(1)}x_{4}^{(4)}\\\
&\quad+x_{1}^{(6)}x_{2}^{(7)}x_{3}^{(1)}x_{4}^{(4)}+x_{1}^{(5)}x_{2}^{(6)}x_{3}^{(4)}x_{4}^{(3)}+x_{1}^{(6)}x_{2}^{(7)}x_{3}^{(2)}x_{4}^{(3)}+x_{1}^{(3)}x_{2}^{(10)}x_{3}^{(2)}x_{4}^{(3)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(3)}x_{2}^{(11)}x_{3}^{(2)}x_{4}^{(2)}\\\
&\quad+x_{1}^{(5)}x_{2}^{(9)}x_{3}^{(2)}x_{4}^{(2)}+x_{1}^{(6)}x_{2}^{(10)}x_{3}^{(1)}x_{4}^{(1)}+x_{1}^{(3)}x_{2}^{(5)}x_{3}^{(9)}x_{4}^{(1)}+x_{1}^{(3)}x_{2}^{(5)}x_{3}^{(8)}x_{4}^{(2)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(3)}x_{2}^{(6)}x_{3}^{(8)}x_{4}^{(1)}\\\
&\quad+x_{1}^{(3)}x_{2}^{(6)}x_{3}^{(7)}x_{4}^{(2)}+x_{1}^{(3)}x_{2}^{(5)}x_{3}^{(6)}x_{4}^{(4)}+x_{1}^{(3)}x_{2}^{(6)}x_{3}^{(6)}x_{4}^{(3)}+x_{1}^{(5)}x_{2}^{(6)}x_{3}^{(6)}x_{4}^{(1)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(3)}x_{2}^{(5)}x_{3}^{(5)}x_{4}^{(5)}\\\
&\quad+x_{1}^{(3)}x_{2}^{(6)}x_{3}^{(5)}x_{4}^{(4)}+x_{1}^{(5)}x_{2}^{(6)}x_{3}^{(5)}x_{4}^{(2)}+x_{1}^{(3)}x_{2}^{(9)}x_{3}^{(5)}x_{4}^{(1)}+x_{1}^{(5)}x_{2}^{(7)}x_{3}^{(5)}x_{4}^{(1)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(3)}x_{2}^{(9)}x_{3}^{(4)}x_{4}^{(2)}\\\
&\quad+x_{1}^{(5)}x_{2}^{(7)}x_{3}^{(4)}x_{4}^{(2)}+x_{1}^{(3)}x_{2}^{(10)}x_{3}^{(4)}x_{4}^{(1)}+x_{1}^{(6)}x_{2}^{(7)}x_{3}^{(4)}x_{4}^{(1)}+x_{1}^{(5)}x_{2}^{(6)}x_{3}^{(3)}x_{4}^{(4)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(6)}x_{2}^{(7)}x_{3}^{(3)}x_{4}^{(2)}\\\
&\quad+x_{1}^{(3)}x_{2}^{(10)}x_{3}^{(3)}x_{4}^{(2)}+x_{1}^{(3)}x_{2}^{(11)}x_{3}^{(2)}x_{4}^{(2)}+x_{1}^{(5)}x_{2}^{(9)}x_{3}^{(2)}x_{4}^{(2)}+x_{1}^{(6)}x_{2}^{(10)}x_{3}^{(1)}x_{4}^{(1)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(3)}x_{2}^{(12)}x_{3}^{(1)}x_{4}^{(2)}\\\
&\quad+x_{1}^{(7)}x_{2}^{(8)}x_{3}^{(1)}x_{4}^{(2)}+x_{1}^{(11)}x_{2}^{(4)}x_{3}^{(1)}x_{4}^{(2)}+x_{1}^{(13)}x_{2}^{(2)}x_{3}^{(1)}x_{4}^{(2)}+x_{1}^{(14)}x_{2}^{(1)}x_{3}^{(1)}x_{4}^{(2)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(12)}x_{2}^{(3)}x_{3}^{(1)}x_{4}^{(2)}\\\
&\quad+x_{1}^{(8)}x_{2}^{(7)}x_{3}^{(1)}x_{4}^{(2)}+x_{1}^{(4)}x_{2}^{(11)}x_{3}^{(1)}x_{4}^{(2)}+x_{1}^{(2)}x_{2}^{(13)}x_{3}^{(1)}x_{4}^{(2)}+x_{1}^{(1)}x_{2}^{(14)}x_{3}^{(1)}x_{4}^{(2)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(6)}x_{2}^{(6)}x_{3}^{(3)}x_{4}^{(3)}\\\
&\quad+x_{1}^{(5)}x_{2}^{(5)}x_{3}^{(5)}x_{4}^{(3)}+x_{1}^{(3)}x_{2}^{(3)}x_{3}^{(9)}x_{4}^{(3)}+x_{1}^{(5)}x_{2}^{(3)}x_{3}^{(7)}x_{4}^{(3)}+x_{1}^{(7)}x_{2}^{(7)}x_{3}^{(2)}x_{4}^{(2)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(6)}x_{2}^{(9)}x_{3}^{(1)}x_{4}^{(2)}\\\
&\quad+x_{1}^{(9)}x_{2}^{(6)}x_{3}^{(1)}x_{4}^{(2)}+x_{1}^{(10)}x_{2}^{(5)}x_{3}^{(1)}x_{4}^{(2)}+x_{1}^{(5)}x_{2}^{(10)}x_{3}^{(2)}x_{4}^{(1)}+x_{1}^{(13)}x_{2}^{(3)}x_{3}^{(1)}x_{4}^{(1)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(5)}x_{2}^{(11)}x_{3}^{(1)}x_{4}^{(1)}\\\
&\quad+x_{1}^{(9)}x_{2}^{(7)}x_{3}^{(1)}x_{4}^{(1)})\in[P_{n_{2}}^{\otimes
4}]^{*}.\end{array}$

Then, it is $\overline{\mathcal{A}}$-annahilated. Indeed, by the unstable
condition, we need only to effects of the Steenrod squares $Sq^{2^{i}}$ for
$i=0,1,2,3.$ It is easy to check that $(\zeta)Sq^{2^{i}}=0$ for $i=0,1,2.$ A
direct computation shows that

$\begin{array}[]{ll}(\zeta)Sq^{8}&=x_{1}^{(3)}x_{2}^{(3)}x_{3}^{(1)}x_{4}^{(3)}+x_{1}^{(3)}x_{2}^{(3)}x_{3}^{(1)}x_{4}^{(3)}+x_{1}^{(3)}x_{2}^{(5)}x_{3}^{(1)}x_{4}^{(1)}+x_{1}^{(3)}x_{2}^{(5)}x_{3}^{(1)}x_{4}^{(1)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(3)}x_{2}^{(3)}x_{3}^{(3)}x_{4}^{(1)}\\\
&+x_{1}^{(3)}x_{2}^{(3)}x_{3}^{(3)}x_{4}^{(1)}+x_{1}^{(3)}x_{2}^{(5)}x_{3}^{(1)}x_{4}^{(1)}+x_{1}^{(3)}x_{2}^{(5)}x_{3}^{(1)}x_{4}^{(1)}+x_{1}^{(7)}x_{2}^{(1)}x_{3}^{(1)}x_{4}^{(1)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(7)}x_{2}^{(1)}x_{3}^{(1)}x_{4}^{(1)}\\\
&+x_{1}^{(1)}x_{2}^{(7)}x_{3}^{(1)}x_{4}^{(1)}+x_{1}^{(1)}x_{2}^{(7)}x_{3}^{(1)}x_{4}^{(1)}+x_{1}^{(3)}x_{2}^{(5)}x_{3}^{(1)}x_{4}^{(1)}+x_{1}^{(3)}x_{2}^{(5)}x_{3}^{(1)}x_{4}^{(1)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(5)}x_{2}^{(3)}x_{3}^{(1)}x_{4}^{(1)}\\\
&+x_{1}^{(5)}x_{2}^{(3)}x_{3}^{(1)}x_{4}^{(1)}=0.\end{array}$

Moreover, it is easy to see that the element
$x_{1}^{(1)}x_{2}^{(1)}x_{3}^{(1)}x_{4}^{(15)}\in[P_{n_{2}}^{\otimes 4}]^{*}$
is also $\overline{\mathcal{A}}$-annahilated. Then, using the representation
in the algebra $\Lambda$ of the fourth cohomological transfer, we deduce that

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt&\psi_{4}(x_{1}^{(1)}x_{2}^{(1)}x_{3}^{(1)}x_{4}^{(15)})=\lambda_{1}^{3}\lambda_{15},\\\
&\psi_{4}(\zeta)=\lambda_{4}\lambda_{6}\lambda_{5}\lambda_{3}+\lambda_{5}\lambda_{7}\lambda^{2}_{3}+\lambda_{3}^{2}\lambda_{2}\lambda_{5}\lambda_{7}+\lambda_{2}\lambda_{4}\lambda_{5}\lambda_{7}+\delta(\lambda_{3}\lambda_{5}\lambda_{11})\end{array}$

are the cycles in $\Lambda^{4,n_{2}}$ and

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt&Tr_{4}^{\mathcal{A}}([x_{1}^{(1)}x_{2}^{(1)}x_{3}^{(1)}x_{4}^{(15)}])=[\psi_{4}(x_{1}^{(1)}x_{2}^{(1)}x_{3}^{(1)}x_{4}^{(15)})]=h_{1}^{3}h_{4}\in{\rm
Ext}_{\mathcal{A}}^{4,4+n_{2}}(\mathbb{Z}/2,\mathbb{Z}/2),\\\
&Tr_{4}^{\mathcal{A}}([\zeta])=[\psi_{4}(\zeta)]=f_{0}\in{\rm
Ext}_{\mathcal{A}}^{4,4+n_{2}}(\mathbb{Z}/2,\mathbb{Z}/2).\end{array}$

Combining the arguments above with the equality (6), we claim that
$\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n_{2}}^{\otimes 4}]^{*}$ is 2-dimensional.

Now, let $\rho\in P^{\otimes 4}_{n_{2}}$ such that $[\rho]\in[QP^{\otimes
4}_{n_{2}}]^{GL_{4}}.$ Since Kameko’s homomorphism

$(\widetilde{Sq^{0}_{*}})_{n_{2}}:QP^{\otimes 4}_{n_{2}}\longrightarrow
QP^{\otimes 4}_{\frac{n_{2}-4}{2}}$

is an epimorphism of $\mathbb{Z}/2GL_{4}$-modules,
$(\widetilde{Sq^{0}_{*}})_{n_{2}}([\rho])$ belongs to the invariant space
$[QP^{\otimes 4}_{\frac{n_{2}-4}{2}}]^{GL_{4}}.$ By Sum [46, Proposition
4.3.2], $[QP^{\otimes
4}_{\frac{n_{2}-4}{2}}]^{GL_{4}}=\langle[\overline{p}_{4,\,2}]\rangle,$ where

$\overline{p}_{4,\,2}=\sum_{1\leq\ell\leq 3}\sum_{1\leq i_{1}\leq\ldots\leq
i_{\ell}\leq 4}t_{i_{1}}t_{i_{2}}^{2}\ldots
t^{2^{\ell-2}}_{i_{\ell-1}}t^{8-2^{\ell-1}}_{i_{\ell}}+t_{1}t_{2}^{2}t_{3}^{2}t_{4}^{2}.$

So,
$(\widetilde{Sq^{0}_{*}})_{n_{2}}([\rho])=\gamma[\varphi(\overline{p}_{4,\,2})]$
where $\gamma\in\mathbb{Z}/2$ and the $\mathbb{Z}/2$-linear map
$\varphi:P^{\otimes 4}_{\frac{n_{2}-4}{2}}\to P^{\otimes 4}_{n_{2}}$ is
determined by $\varphi(u)=t_{1}t_{2}t_{3}t_{4}u^{2}$ for any $u\in P^{\otimes
4}_{\frac{n_{2}-4}{2}}.$ Because $[\rho]\in[QP^{\otimes 4}_{n_{2}}]^{GL_{4}},$
we have $\rho\equiv\gamma\varphi(\overline{p}_{4,\,2})+\rho^{*},$ where
$\rho^{*}\in P^{\otimes 4}_{n_{2}}$ such that $[\rho^{*}]\in{\rm
Ker}_{n_{2}}.$ By a direct computation using Proposition 3.1.4 and the
relations $\theta_{i}(\rho)+\rho\equiv 0$ for $1\leq i\leq 4$, we get
$[QP^{\otimes
4}_{n_{2}}]^{GL_{4}}=\langle[\varphi(\overline{p}_{4,\,2})],[\widehat{p_{2,\,1}}]\rangle$.
Furthermore, we see that

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt&\langle[\varphi(\overline{p}_{4,\,2})],[\zeta]\rangle=0,\ \
\langle[\widehat{p_{2,\,1}})],[x_{1}^{(1)}x_{2}^{(1)}x_{3}^{(1)}x_{4}^{(15)}]\rangle=0,\\\
&\langle[\widehat{p_{2,\,1}}],[\zeta]\rangle=1,\ \ \ \ \ \
\langle[\varphi(\overline{p}_{4,\,2})],[x_{1}^{(1)}x_{2}^{(1)}x_{3}^{(1)}x_{4}^{(15)}]\rangle=1.\end{array}$

Therefore, $\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n_{2}}^{\otimes
4}]^{*}=\langle[x_{1}^{(1)}x_{2}^{1}x_{3}^{(1)}x_{4}^{(15)}],[\zeta]\rangle.$
The theorem is proved. ∎

###### Proof of Theorem 2.1.6.

We consider the following elements in $[P_{n_{3}}^{\otimes 4}]^{*}$:
$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt&\overline{\zeta}=x_{1}^{(0)}x_{2}^{(0)}x_{3}^{(7)}x_{4}^{(31)},\\\
&\zeta=x_{1}^{(11)}x_{2}^{(11)}x_{3}^{(11)}x_{4}^{(5)}+x_{1}^{(11)}x_{2}^{(11)}x_{3}^{(13)}x_{4}^{(3)}+x_{1}^{(7)}x_{2}^{(11)}x_{3}^{(17)}x_{4}^{(3)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(11)}x_{2}^{(7)}x_{3}^{(17)}x_{4}^{(3)}\\\
&+x_{1}^{(11)}x_{2}^{(15)}x_{3}^{(9)}x_{4}^{(3)}+x_{1}^{(15)}x_{2}^{(11)}x_{3}^{(9)}x_{4}^{(3)}+x_{1}^{(7)}x_{2}^{(19)}x_{3}^{(9)}x_{4}^{(3)}+x_{1}^{(19)}x_{2}^{(7)}x_{3}^{(9)}x_{4}^{(3)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(7)}x_{2}^{(19)}x_{3}^{(7)}x_{4}^{(5)}\\\
&+x_{1}^{(19)}x_{2}^{(7)}x_{3}^{(7)}x_{4}^{(5)}+x_{1}^{(11)}x_{2}^{(19)}x_{3}^{(5)}x_{4}^{(3)}+x_{1}^{(19)}x_{2}^{(11)}x_{3}^{(5)}x_{4}^{(3)}+x_{1}^{(11)}x_{2}^{(21)}x_{3}^{(3)}x_{4}^{(3)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(19)}x_{2}^{(13)}x_{3}^{(3)}x_{4}^{(3)}\\\
&+x_{1}^{(7)}x_{2}^{(23)}x_{3}^{(5)}x_{4}^{(3)}+x_{1}^{(23)}x_{2}^{(7)}x_{3}^{(5)}x_{4}^{(3)}+x_{1}^{(11)}x_{2}^{(11)}x_{3}^{(7)}x_{4}^{(9)}+x_{1}^{(11)}x_{2}^{(7)}x_{3}^{(11)}x_{4}^{(9)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(7)}x_{2}^{(11)}x_{3}^{(11)}x_{4}^{(9)}\\\
&+x_{1}^{(7)}x_{2}^{(25)}x_{3}^{(3)}x_{4}^{(3)}+x_{1}^{(23)}x_{2}^{(9)}x_{3}^{(3)}x_{4}^{(3)}+x_{1}^{(15)}x_{2}^{(17)}x_{3}^{(3)}x_{4}^{(3)}+x_{1}^{(15)}x_{2}^{(15)}x_{3}^{(3)}x_{4}^{(5)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(27)}x_{2}^{(5)}x_{3}^{(3)}x_{4}^{(3)}\\\
&+x_{1}^{(29)}x_{2}^{(3)}x_{3}^{(3)}x_{4}^{(3)}+x_{1}^{(13)}x_{2}^{(11)}x_{3}^{(7)}x_{4}^{(7)}+x_{1}^{(11)}x_{2}^{(7)}x_{3}^{(13)}x_{4}^{(7)}+x_{1}^{(7)}x_{2}^{(13)}x_{3}^{(11)}x_{4}^{(7)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(13)}x_{2}^{(7)}x_{3}^{(7)}x_{4}^{(11)}\\\
&+x_{1}^{(7)}x_{2}^{(7)}x_{3}^{(13)}x_{4}^{(11)}+x_{1}^{(7)}x_{2}^{(13)}x_{3}^{(7)}x_{4}^{(11)}+x_{1}^{(11)}x_{2}^{(7)}x_{3}^{(7)}x_{4}^{(13)}+x_{1}^{(7)}x_{2}^{(11)}x_{3}^{(7)}x_{4}^{(13)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(7)}x_{2}^{(7)}x_{3}^{(11)}x_{4}^{(13)}\\\
&+x_{1}^{(7)}x_{2}^{(7)}x_{3}^{(7)}x_{4}^{(17)}+x_{1}^{(7)}x_{2}^{(7)}x_{3}^{(9)}x_{4}^{(15)}+x_{1}^{(7)}x_{2}^{(11)}x_{3}^{(5)}x_{4}^{(15)}+x_{1}^{(7)}x_{2}^{(13)}x_{3}^{(3)}x_{4}^{(15)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(7)}x_{2}^{(7)}x_{3}^{(19)}x_{4}^{(5)}\\\
&+x_{1}^{(7)}x_{2}^{(7)}x_{3}^{(21)}x_{4}^{(3)}+x_{1}^{(11)}x_{2}^{(7)}x_{3}^{(15)}x_{4}^{(5)}+x_{1}^{(11)}x_{2}^{(15)}x_{3}^{(7)}x_{4}^{(5)}+x_{1}^{(15)}x_{2}^{(11)}x_{3}^{(7)}x_{4}^{(5)}+x_{1}^{(7)}x_{2}^{(13)}x_{3}^{(15)}x_{4}^{(3)}.\end{array}$
Then, we can immediately see that $\overline{\zeta}\in{\rm
Ext}_{\mathcal{A}}^{0,n_{3}}(\mathbb{Z}/2,P^{\otimes 4}).$ Further, by direct
calculations, we get $(\zeta)Sq^{2^{i}}=0$ for $0\leq i\leq 2,$ and
$\begin{array}[]{ll}(\zeta)Sq^{8}&=x_{1}^{(7)}x_{2}^{(7)}x_{3}^{(11)}x_{4}^{(5)}+x_{1}^{(7)}x_{2}^{(11)}x_{3}^{(7)}x_{4}^{(5)}+x_{1}^{(11)}x_{2}^{(7)}x_{3}^{(7)}x_{4}^{(5)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(7)}x_{2}^{(7)}x_{3}^{(13)}x_{4}^{(3)}\\\
&+x_{1}^{(7)}x_{2}^{(7)}x_{3}^{(13)}x_{4}^{(3)}+x_{1}^{(11)}x_{2}^{(7)}x_{3}^{(9)}x_{4}^{(3)}+x_{1}^{(7)}x_{2}^{(7)}x_{3}^{(13)}x_{4}^{(3)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(7)}x_{2}^{(15)}x_{3}^{(5)}x_{4}^{(3)}\\\
&+x_{1}^{(15)}x_{2}^{(7)}x_{3}^{(5)}x_{4}^{(3)}+x_{1}^{(7)}x_{2}^{(11)}x_{3}^{(9)}x_{4}^{(3)}+x_{1}^{(7)}x_{2}^{(15)}x_{3}^{(5)}x_{4}^{(3)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(11)}x_{2}^{(7)}x_{3}^{(9)}x_{4}^{(3)}\\\
&+x_{1}^{(15)}x_{2}^{(7)}x_{3}^{(5)}x_{4}^{(3)}+x_{1}^{(7)}x_{2}^{(11)}x_{3}^{(7)}x_{4}^{(5)}+x_{1}^{(11)}x_{2}^{(7)}x_{3}^{(7)}x_{4}^{(5)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(11)}x_{2}^{(11)}x_{3}^{(5)}x_{4}^{(3)}\\\
\end{array}$

$\begin{array}[]{ll}&+x_{1}^{(7)}x_{2}^{(15)}x_{3}^{(5)}x_{4}^{(3)}+x_{1}^{(11)}x_{2}^{(11)}x_{3}^{(5)}x_{4}^{(3)}+x_{1}^{(15)}x_{2}^{(7)}x_{3}^{(5)}x_{4}^{(3)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(11)}x_{2}^{(13)}x_{3}^{(3)}x_{4}^{(3)}\\\
&+x_{1}^{(11)}x_{2}^{(13)}x_{3}^{(3)}x_{4}^{(3)}+x_{1}^{(7)}x_{2}^{(15)}x_{3}^{(5)}x_{4}^{(3)}+x_{1}^{(15)}x_{2}^{(7)}x_{3}^{(5)}x_{4}^{(3)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(7)}x_{2}^{(7)}x_{3}^{(7)}x_{4}^{(9)}\\\
&+x_{1}^{(7)}x_{2}^{(11)}x_{3}^{(7)}x_{4}^{(5)}+x_{1}^{(11)}x_{2}^{(7)}x_{3}^{(7)}x_{4}^{(5)}+x_{1}^{(7)}x_{2}^{(7)}x_{3}^{(7)}x_{4}^{(9)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(7)}x_{2}^{(7)}x_{3}^{(11)}x_{4}^{(5)}\\\
&+x_{1}^{(11)}x_{2}^{(7)}x_{3}^{(7)}x_{4}^{(5)}+x_{1}^{(7)}x_{2}^{(7)}x_{3}^{(7)}x_{4}^{(9)}+x_{1}^{(7)}x_{2}^{(7)}x_{3}^{(11)}x_{4}^{(5)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(7)}x_{2}^{(11)}x_{3}^{(7)}x_{4}^{(5)}\\\
&+x_{1}^{(15)}x_{2}^{(9)}x_{3}^{(3)}x_{4}^{(3)}+x_{1}^{(15)}x_{2}^{(9)}x_{3}^{(3)}x_{4}^{(3)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(7)}x_{2}^{(7)}x_{3}^{(7)}x_{4}^{(9)}\\\
&+x_{1}^{(7)}x_{2}^{(7)}x_{3}^{(11)}x_{4}^{(5)}+x_{1}^{(7)}x_{2}^{(7)}x_{3}^{(13)}x_{4}^{(3)}+x_{1}^{(7)}x_{2}^{(11)}x_{3}^{(9)}x_{4}^{(3)}=0.\end{array}$
Thus, by the unstable condition, we claim that $\zeta\in{\rm
Ext}_{\mathcal{A}}^{0,n_{3}}(\mathbb{Z}/2,P^{\otimes 4}).$ Using the
representation of the rank 4 algebraic transfer over the algebra $\Lambda$, we
find that the following cycles

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\psi_{4}(\overline{\zeta})&=\lambda_{0}^{2}\lambda_{7}\lambda_{31}\in\Lambda^{4,n_{3}}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\psi_{4}(\zeta)&=(\lambda_{7}^{3}\lambda_{17}+(\lambda_{7}\lambda_{11}^{2}+\lambda_{7}^{2}\lambda_{15})\lambda_{9}+\lambda_{15}\lambda_{11}\lambda_{7}\lambda_{5}+\lambda_{7}^{2}\lambda_{11}\lambda_{13}\\\
&\quad+\delta(\lambda_{7}\lambda_{11}\lambda_{21}+\lambda_{7}\lambda_{25}\lambda_{7}+\lambda_{9}\lambda_{15}^{2}+\lambda_{1}\lambda_{23}\lambda_{15}))\in\Lambda^{4,n_{3}}\end{array}$

in $\Lambda$ are representative of the non-zero elements $h_{0}^{2}h_{3}h_{5}$
and $e_{1}\in{\rm Ext}_{\mathcal{A}}^{4,4+n_{3}}(\mathbb{Z}/2,\mathbb{Z}/2)$
respectively. So, by the equality (6), we get
$\dim\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n_{3}}^{\otimes 4}]^{*}\geq 2.$ On the other
hand, we have shown in [38] that $\dim\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{\frac{n_{3}-4}{2}}^{\otimes 4}]^{*}=1.$
Combining these data with the inequality (5) and Proposition 3.1.4, we obtain
$\dim\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n_{3}}^{\otimes 4}]^{*}=2.$

Now, if $[\widetilde{\rho}]\in[QP^{\otimes 4}_{n_{3}}]^{GL_{4}},$ then the
invariant space $[QP^{\otimes 4}_{\frac{n_{3}-4}{2}}]^{GL_{4}}$ contains the
image of $[\widetilde{\rho}]$ under Kameko’s homomorphism
$(\widetilde{Sq^{0}_{*}})_{n_{3}}.$ On the other hand, in [38], we showed that
$[QP^{\otimes
4}_{\frac{n_{3}-4}{2}}]^{GL_{4}}=\langle[\widetilde{\zeta}]\rangle$ with

$\begin{array}[]{ll}\widetilde{\zeta}&=t_{1}t_{2}^{2}t_{3}^{7}t_{4}^{7}+t_{1}t_{2}^{7}t_{3}^{2}t_{4}^{7}+t_{1}t_{2}^{3}t_{3}t_{4}^{12}+\vskip
6.0pt plus 2.0pt minus 2.0ptt_{1}t_{2}^{3}t_{3}^{12}t_{4}\\\
&\quad+t_{1}^{3}t_{2}t_{3}t_{4}^{12}+t_{1}^{3}t_{2}t_{3}^{12}t_{4}+t_{1}^{3}t_{2}^{5}t_{3}t_{4}^{8}+t_{1}^{3}t_{2}^{5}t_{3}^{8}t_{4}.\end{array}$

Hence,
$(\widetilde{Sq^{0}_{*}})_{n_{3}}([\widetilde{\rho}])=\beta[\overline{\varphi}(\widetilde{\zeta})]$
where $\beta\in\mathbb{Z}/2$ and the $\mathbb{Z}/2$-linear map
$\overline{\varphi}:P^{\otimes 4}_{\frac{n_{3}-4}{2}}\to P^{\otimes
4}_{n_{3}}$ is a linear map determined by $\overline{\varphi}(v)=\prod_{1\leq
i\leq 4}t_{i}v^{2}$ for all $v=\prod_{1\leq i\leq 4}t_{i}^{a_{i}}\in
P^{\otimes 4}_{\frac{n_{3}-4}{2}}.$ Thus, we have
$\widetilde{\rho}\equiv\beta\overline{\varphi}(\widetilde{\zeta})+\overline{\rho},$
where $\overline{\rho}\in P^{\otimes 4}_{n_{3}}$ such that
$[\overline{\rho}]\in{\rm Ker}_{n_{3}}.$ Using Proposition 3.1.4 and the
relations $\theta_{i}(\widetilde{\rho})\equiv\widetilde{\rho},\,i=1,2,3,4,$ we
obtain
$\widetilde{\rho}\equiv\beta\overline{\varphi}(\widetilde{\zeta})+\beta^{\prime}\sum_{1\leq
j\leq 105}{\rm adm}_{3,\,j},\ \ (\beta^{\prime}\in\mathbb{Z}/2).$ Combining
this and the fact that

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt&\langle[\overline{\varphi}(\widetilde{\zeta})],[\overline{\zeta}]\rangle=0,\
\ \langle[\sum_{1\leq j\leq 105}{\rm adm}_{3,\,j}],[\zeta]\rangle=0,\\\
&\langle[\sum_{1\leq j\leq 105}{\rm
adm}_{3,\,j}],[\overline{\zeta}]\rangle=1,\ \
\langle[\overline{\varphi}(\widetilde{\zeta})],[\zeta]\rangle=1,\end{array}$

we get $\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n_{3}}^{\otimes
4}]^{*}=\langle[\overline{\zeta}],[\zeta]\rangle.$ The proof of the theorem is
completed. ∎

###### Proof of Theorem 2.1.8.

From the inequality (5), Proposition 3.1.4 and our result in [38] that the
coinvarian space $\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{\frac{n_{s}-4}{2}}^{\otimes 4}]^{*}$ is
trivial for all $s\geq 4$, one gets an estimate that
$\dim\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n_{3}}^{\otimes 4}]^{*}\leq 1.$ On the other
hand, it is easy to see that the element
$\zeta_{s}=x_{1}^{(0)}x_{2}^{(0)}x_{3}^{(2^{s}-1)}x_{4}^{(2^{s+2}-1)}$ belongs
to ${\rm Ext}_{\mathcal{A}}^{0,n_{s}}(\mathbb{Z}/2,P^{\otimes 4}).$ So, based
on the equality (6) and a representation of the rank 4 transfer over the
lambda algebra, we claim that
$h_{0}^{2}h_{s}h_{s+2}=Tr_{4}^{\mathcal{A}}([\zeta])=[\psi_{4}(\zeta)]=[\lambda_{0}^{2}\lambda_{2^{s}-1}\lambda_{2^{s+2}-1}]\in{\rm
Ext}_{\mathcal{A}}^{4,4+n_{s}}(\mathbb{Z}/2,\mathbb{Z}/2)$ and
$\dim\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n_{s}}^{\otimes 4}]^{*}\geq 1.$ It should be
noted that
$\lambda_{0}^{2}\lambda_{2^{s}-1}\lambda_{2^{s+2}-1}\in\Lambda^{4,n_{s}}$ is a
cycle in $\Lambda.$ Thus, $\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n_{3}}^{\otimes 4}]^{*}$ is 1-dimensional.
Furthermore, because $\langle[\sum_{1\leq j\leq 105}{\rm
adm}_{s,\,j}],[\zeta_{s}]\rangle=1,$ for any $s>3,$ by Proposition 3.1.4, we
conclude that the set $\\{[\zeta_{s}]\\}$ is a basis of
$\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n_{s}}^{\otimes 4}]^{*}.$ This completes the
proof of the theorem. ∎

###  3.2 The degree $n^{\prime}_{s}=17.2^{s}-2$

In this section, we prove Theorems 2.2.1, 2.2.3 and 2.2.5. Firstly, we need
the following results. Let us denote ${\rm Ker}_{n^{\prime}_{s}}$ the kernel
of Kameko’s homomorphism
$(\widetilde{Sq^{0}_{*}})_{n^{\prime}_{s}}:QP^{\otimes
4}_{n^{\prime}_{s}}\longrightarrow QP^{\otimes
4}_{\frac{n^{\prime}_{s}-4}{2}}.$ According to Sum [43, Propositions 7.2.1,
7.4.1, Theorems 7.2.2, 7.4.2]), ${\rm Ker}_{n^{\prime}_{s}}$ has a basis
consisting all the classes represented by the admissible monomials
$b_{4,\,1,\,j},$ $c_{4,\,1,\,j}$ ($s=1$), and $b_{4,\,s,\,j}$ ($s\geq 2).$
Based on this result, we obtain the following.

######  Proposition 3.2.1.

Let $s$ be a positive integer. Then,

$[{\rm
Ker}_{n^{\prime}_{s}}]^{GL_{4}}=\left\\{\begin{array}[]{ll}\langle[\sum_{1\leq
j\leq 105}b_{4,\,s,\,j}]\rangle&\mbox{if $s=3$},\\\ 0&\mbox{if $s\neq
3$}.\end{array}\right.$

###### Proof.

The proof of the proposition for the cases $s\geq 2$ is similar to Proposition
3.1.4. Now, we prove the case $s=1$ in detail. As well known (see Sum [43]),
an admissible monomial basis for ${\rm Ker}_{n^{\prime}_{1}}$ is the set
$\\{[{\rm adm}_{j}]:\,1\leq j\leq 45\\}=\\{[b_{4,\,1,\,j}]:\,1\leq j\leq
38\\}\bigcup\\{[c_{4,\,1,\,j}]:\,1\leq j\leq 7\\},$ where

${\rm adm}_{1}=t_{3}t_{4}^{31}$, |  ${\rm adm}_{2}=t_{3}^{31}t_{4}$, |  ${\rm adm}_{3}=t_{2}t_{4}^{31}$, |  ${\rm adm}_{4}=t_{2}t_{3}^{31}$,
---|---|---|---
${\rm adm}_{5}=t_{2}^{31}t_{4}$, |  ${\rm adm}_{6}=t_{2}^{31}t_{3}$, |  ${\rm adm}_{7}=t_{1}t_{4}^{31}$, |  ${\rm adm}_{8}=t_{1}t_{3}^{31}$,
${\rm adm}_{9}=t_{1}t_{2}^{31}$, |  ${\rm adm}_{10}=t_{1}^{31}t_{4}$, |  ${\rm adm}_{11}=t_{1}^{31}t_{3}$, |  ${\rm adm}_{12}=t_{1}^{31}t_{2}$,
${\rm adm}_{13}=t_{3}^{3}t_{4}^{29}$, |  ${\rm adm}_{14}=t_{2}^{3}t_{4}^{29}$, |  ${\rm adm}_{15}=t_{2}^{3}t_{3}^{29}$, |  ${\rm adm}_{16}=t_{1}^{3}t_{4}^{29}$,
${\rm adm}_{17}=t_{1}^{3}t_{3}^{29}$, |  ${\rm adm}_{18}=t_{1}^{3}t_{2}^{29}$, |  ${\rm adm}_{19}=t_{2}t_{3}t_{4}^{30}$, |  ${\rm adm}_{20}=t_{2}t_{3}^{30}t_{4}$,
${\rm adm}_{21}=t_{1}t_{3}t_{4}^{30}$, |  ${\rm adm}_{22}=t_{1}t_{3}^{30}t_{4}$, |  ${\rm adm}_{23}=t_{1}t_{2}t_{4}^{30}$, |  ${\rm adm}_{24}=t_{1}t_{2}t_{3}^{30}$,
${\rm adm}_{25}=t_{1}t_{2}^{30}t_{4}$, |  ${\rm adm}_{26}=t_{1}t_{2}^{30}t_{3}$, |  ${\rm adm}_{27}=t_{2}t_{3}^{2}t_{4}^{29}$, |  ${\rm adm}_{28}=t_{1}t_{3}^{2}t_{4}^{29}$,
${\rm adm}_{29}=t_{1}t_{2}^{2}t_{4}^{29}$, |  ${\rm adm}_{30}=t_{1}t_{2}^{2}t_{3}^{29}$, |  ${\rm adm}_{31}=t_{2}t_{3}^{3}t_{4}^{28}$, |  ${\rm adm}_{32}=t_{2}^{3}t_{3}t_{4}^{28}$,
${\rm adm}_{33}=t_{1}t_{3}^{3}t_{4}^{28}$, |  ${\rm adm}_{34}=t_{1}t_{2}^{3}t_{4}^{28}$, |  ${\rm adm}_{35}=t_{1}t_{2}^{3}t_{3}^{28}$, |  ${\rm adm}_{36}=t_{1}^{3}t_{3}t_{4}^{28}$,
${\rm adm}_{37}=t_{1}^{3}t_{2}t_{4}^{28}$, |  ${\rm adm}_{38}=t_{1}^{3}t_{2}t_{3}^{28}$, |  ${\rm adm}_{39}=t_{1}t_{2}t_{3}^{2}t_{4}^{28}$, |  ${\rm adm}_{40}=t_{1}t_{2}^{2}t_{3}t_{4}^{28}$,
${\rm adm}_{41}=t_{1}t_{2}^{2}t_{3}^{28}t_{4}$, |  ${\rm adm}_{42}=t_{1}t_{2}^{2}t_{3}^{4}t_{4}^{25}$, |  ${\rm adm}_{43}=t_{1}t_{2}^{2}t_{3}^{5}t_{4}^{24}$, |  ${\rm adm}_{44}=t_{1}t_{2}^{3}t_{3}^{4}t_{4}^{24}$,
${\rm adm}_{45}=t_{1}^{3}t_{2}t_{3}^{4}t_{4}^{24}$. |  |  |

Then we have a direct summand decomposition of $S_{4}$-submodules: ${\rm
Ker}_{n^{\prime}_{1}}=\bigoplus_{1\leq i\leq 4}\mathbb{M}_{i},$ where

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt&\mathbb{M}_{1}:=S_{4}({\rm adm}_{1})=\langle\\{[{\rm adm}_{j}]:\,1\leq
j\leq 12\\}\rangle,\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt&\mathbb{M}_{2}:=S_{4}({\rm adm}_{13})=\langle\\{[{\rm adm}_{j}]:\,13\leq
j\leq 18\\}\rangle,\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt&\mathbb{M}_{3}:=S_{4}({\rm adm}_{19},{\rm adm}_{27},{\rm
adm}_{31})=\langle\\{[{\rm adm}_{j}]:\,19\leq j\leq 38\\}\rangle,\\\ \vskip
6.0pt plus 2.0pt minus 2.0pt&\mathbb{M}_{4}:=\langle\\{[{\rm
adm}_{j}]:\,39\leq j\leq 45\\}\rangle.\end{array}$

Denote by $\mathscr{B}_{i}$ the bases of the spaces $\mathbb{M}_{i}.$ Let
$f_{i}\in P^{\otimes 4}_{n^{\prime}_{1}}$ such that
$[f_{i}]\in\mathbb{M}_{i}^{S_{4}},$ then
$f_{i}\equiv\sum_{u\in\mathscr{B}_{i}}\gamma_{u}u$ with
$\gamma_{u}\in\mathbb{Z}/2.$ For each $1\leq k\leq 3,$ we compute explicitly
the equalities $\theta_{k}(f_{i})$ in the terms $u\in\mathscr{B}_{i}$ modulo
($\overline{\mathcal{A}}P^{\otimes 4}_{n^{\prime}_{1}}$). Then, from the
relations $\theta_{k}(f_{i})+f_{i}\equiv 0$, we find that

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt&\mathbb{M}_{1}^{S_{4}}=\langle[q_{1}]\rangle\ \mbox{with\
$q_{1}=\sum_{1\leq j\leq 12}{\rm adm}_{j}$},\ \
\mathbb{M}_{2}^{S_{4}}=\langle[q_{2}]\rangle\ \mbox{with\ $q_{2}=\sum_{13\leq
j\leq 18}{\rm adm}_{j}$},\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt&\mathbb{M}_{3}^{S_{4}}=\langle[q_{3}]\rangle\ \mbox{with\ $q_{3}={\rm
adm}_{20}+{\rm adm}_{22}+\sum_{25\leq j\leq 38}{\rm adm}_{j}$},\\\ \vskip
6.0pt plus 2.0pt minus
2.0pt&\mathbb{M}_{4}^{S_{4}}=\langle[q_{4}+q_{5}],[q_{5}+q_{6}]\rangle\
\mbox{with\ $q_{4}=\sum_{42\leq j\leq 45}{\rm adm}_{j}$, $q_{5}={\rm
adm}_{39}$, $q_{6}={\rm adm}_{40}+{\rm adm}_{41}$}.\end{array}$

Now, suppose that $[h]\in[{\rm Ker}_{n^{\prime}_{s}}]^{GL_{4}},$ then because
$S_{4}\subset GL_{4},$ we have

$h\equiv\xi_{1}q_{1}+\xi_{2}q_{2}+\xi_{3}q_{3}+\xi_{4}(q_{4}+q_{5})+\xi_{5}(q_{5}+q_{6}),$

in which $\xi_{i}\in\mathbb{Z}/2.$ A direct computation shows that

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\theta_{4}(h)+h&\equiv(\xi_{1}{\rm adm}_{3}+(\xi_{1}+\xi_{3}){\rm
adm}_{5}+(\xi_{1}+\xi_{2}){\rm adm}_{9}+(\xi_{3}+\xi_{4}){\rm adm}_{27}\\\
&\qquad+(\xi_{3}+\xi_{5}){\rm adm}_{20}+\ \mbox{other terms})\ \equiv
0.\end{array}$

This equality implies that $\xi_{i}=0$ for $1\leq i\leq 5.$ The proposition
follows. ∎

######  Remark 3.2.2.

From a result of Lin [20], we claim that

(8) ${\rm
Ext}_{\mathcal{A}}^{4,4+n^{\prime}_{s}}(\mathbb{Z}/2,\mathbb{Z}/2)=\left\\{\begin{array}[]{ll}\langle
d_{1}\rangle&\mbox{if $s=1$},\\\\[2.84526pt] \langle
h_{1}^{3}h_{6}\rangle=\langle h_{0}^{2}h_{2}h_{6}\rangle&\mbox{if
$s=2$},\\\\[2.84526pt] \langle
h_{0}^{2}h_{3}h_{7},h_{1}h_{3}h_{6}^{2}\rangle&\mbox{if $s=3$},\\\\[2.84526pt]
\langle h_{1}h^{2}_{s-1}h_{s+4},h_{1}h_{s}h_{s+3}^{2}\rangle&\mbox{if $s\geq
4$},\end{array}\right.$

where $h_{1}h^{2}_{s-1}h_{s+4}=0$ for $s=4$ and $h_{1}h^{2}_{s-1}h_{s+4}\neq
h_{1}h_{s}h_{s+3}^{2}$ for $s\geq 5.$

###### Proof of Theorem 2.2.1.

By Sum [46], the coinvariant space $\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{\frac{n^{\prime}_{1}-4}{2}}^{\otimes 4}]^{*}$
has dimension 1. Combining this with the inequality (7) and Proposition 3.2.1,
we deduce an estimate that $\dim\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n^{\prime}_{1}}^{\otimes 4}]^{*}\leq 1.$ On
the other hand, it is easy to verify that the element

$\begin{array}[]{ll}\overline{\zeta}&=x_{1}^{(3)}x_{2}^{(13)}x_{3}^{(7)}x_{4}^{(9)}+x_{1}^{(3)}x_{2}^{(13)}x_{3}^{(11)}x_{4}^{(5)}+x_{1}^{(3)}x_{2}^{(13)}x_{3}^{(13)}x_{4}^{(3)}+x_{1}^{(5)}x_{2}^{(11)}x_{3}^{(7)}x_{4}^{(9)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(5)}x_{2}^{(11)}x_{3}^{(11)}x_{4}^{(5)}\\\
&+x_{1}^{(5)}x_{2}^{(11)}x_{3}^{(13)}x_{4}^{(3)}+x_{1}^{(5)}x_{2}^{(13)}x_{3}^{(7)}x_{4}^{(7)}+x_{1}^{(7)}x_{2}^{(3)}x_{3}^{(11)}x_{4}^{(11)}+x_{1}^{(7)}x_{2}^{(3)}x_{3}^{(13)}x_{4}^{(9)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(7)}x_{2}^{(5)}x_{3}^{(11)}x_{4}^{(9)}\\\
&+x_{1}^{(7)}x_{2}^{(5)}x_{3}^{(13)}x_{4}^{(7)}+x_{1}^{(7)}x_{2}^{(7)}x_{3}^{(7)}x_{4}^{(11)}+x_{1}^{(7)}x_{2}^{(7)}x_{3}^{(9)}x_{4}^{(9)}+x_{1}^{(7)}x_{2}^{(7)}x_{3}^{(13)}x_{4}^{(5)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(7)}x_{2}^{(9)}x_{3}^{(7)}x_{4}^{(9)}\\\
&+x_{1}^{(7)}x_{2}^{(11)}x_{3}^{(5)}x_{4}^{(9)}+x_{1}^{(7)}x_{2}^{(13)}x_{3}^{(3)}x_{4}^{(9)}+x_{1}^{(7)}x_{2}^{(13)}x_{3}^{(5)}x_{4}^{(7)}+x_{1}^{(9)}x_{2}^{(7)}x_{3}^{(7)}x_{4}^{(9)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(9)}x_{2}^{(7)}x_{3}^{(11)}x_{4}^{(5)}\\\
&+x_{1}^{(9)}x_{2}^{(7)}x_{3}^{(13)}x_{4}^{(3)}+x_{1}^{(11)}x_{2}^{(3)}x_{3}^{(7)}x_{4}^{(11)}+x_{1}^{(11)}x_{2}^{(3)}x_{3}^{(13)}x_{4}^{(5)}+x_{1}^{(11)}x_{2}^{(5)}x_{3}^{(11)}x_{4}^{(5)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(11)}x_{2}^{(7)}x_{3}^{(3)}x_{4}^{(11)}\\\
&+x_{1}^{(11)}x_{2}^{(7)}x_{3}^{(9)}x_{4}^{(5)}+x_{1}^{(11)}x_{2}^{(9)}x_{3}^{(7)}x_{4}^{(5)}+x_{1}^{(11)}x_{2}^{(11)}x_{3}^{(3)}x_{4}^{(7)}+x_{1}^{(11)}x_{2}^{(11)}x_{3}^{(5)}x_{4}^{(5)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(11)}x_{2}^{(13)}x_{3}^{(3)}x_{4}^{(5)}\\\
&+x_{1}^{(13)}x_{2}^{(3)}x_{3}^{(13)}x_{4}^{(3)}+x_{1}^{(13)}x_{2}^{(5)}x_{3}^{(11)}x_{4}^{(3)}+x_{1}^{(13)}x_{2}^{(7)}x_{3}^{(7)}x_{4}^{(5)}+x_{1}^{(13)}x_{2}^{(7)}x_{3}^{(7)}x_{4}^{(5)}+\vskip
6.0pt plus 2.0pt minus 2.0ptx_{1}^{(13)}x_{2}^{(7)}x_{3}^{(9)}x_{4}^{(3)}\\\
&+x_{1}^{(13)}x_{2}^{(9)}x_{3}^{(7)}x_{4}^{(3)}+x_{1}^{(13)}x_{2}^{(11)}x_{3}^{(5)}x_{4}^{(3)}+x_{1}^{(13)}x_{2}^{(13)}x_{3}^{(3)}x_{4}^{(3)}\end{array}$

belongs to ${\rm Ext}_{\mathcal{A}}^{0,n^{\prime}_{1}}(\mathbb{Z}/2,P^{\otimes
4}).$ It should be note that we need only to consider effects of the Steenrod
squares $Sq^{i}$ for $i=1,2,4,8$ because of the unstable condition. So, using
the representation of $Tr_{4}^{\mathcal{A}}$ over $\Lambda$, we find that
$\psi_{4}(\overline{\zeta})=\lambda_{7}^{2}\lambda_{5}\lambda_{13}+\lambda_{7}^{2}\lambda_{9}^{2}+\lambda_{7}\lambda_{11}\lambda_{9}\lambda_{5}+\lambda_{15}\lambda_{3}\lambda_{11}\lambda_{3}+\delta(\lambda_{7}^{2}\lambda_{19}+\lambda_{7}\lambda_{19}\lambda_{7})$
is a cycle in $\Lambda$ and is a representative of the non-zero element
$d_{1}\in{\rm
Ext}_{\mathcal{A}}^{4,4+n^{\prime}_{1}}(\mathbb{Z}/2,\mathbb{Z}/2).$ Combining
this with the equality (8) and the fact that
$\dim\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n^{\prime}_{1}}^{\otimes 4}]^{*}\leq 1$, we
claim that the coinvariant space $\dim\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n^{\prime}_{1}}^{\otimes 4}]^{*}$ is
1-dimensional. Now, let $\sigma\in P^{\otimes 4}_{n^{\prime}_{1}}$ such that
$[\sigma]\in[QP^{\otimes 4}_{n^{\prime}_{1}}]^{GL_{4}}.$ Consider the linear
transformation

$\begin{array}[]{ll}\widetilde{\varphi}:P^{\otimes
4}_{\frac{n^{\prime}_{1}-4}{2}}&\longrightarrow P^{\otimes
4}_{n^{\prime}_{1}},\\\ \prod_{1\leq i\leq
4}t_{i}^{a_{i}}&\longmapsto\prod_{1\leq i\leq 4}t_{i}^{2a_{i}+1}.\end{array}$

According to Sum [46, Proposition 4.2.2], the invariant space $[QP^{\otimes
4}_{\frac{n^{\prime}_{1}-4}{2}}]^{GL_{4}}$ is generated by the class
$[t_{1}t_{2}t_{3}^{6}t_{4}^{6}+t_{1}^{3}t_{2}^{3}t_{3}^{4}t_{4}^{4}].$ Then,
since the Kameko squaring operation
$(\widetilde{Sq^{0}_{*}})_{n^{\prime}_{1}}$ is an epimorphism of
$\mathbb{Z}/2GL_{4}$-modules,
$(\widetilde{Sq^{0}_{*}})_{n^{\prime}_{1}}([\sigma])=\gamma[\widetilde{\varphi}(t_{1}t_{2}t_{3}^{6}t_{4}^{6}+t_{1}^{3}t_{2}^{3}t_{3}^{4}t_{4}^{4})]$
with $\gamma\in\mathbb{Z}/2.$ We therefore deduce that
$\sigma\equiv\gamma\widetilde{\varphi}(t_{1}t_{2}t_{3}^{6}t_{4}^{6}+t_{1}^{3}t_{2}^{3}t_{3}^{4}t_{4}^{4})+\sigma^{*},$
where $\sigma^{*}\in P^{\otimes 4}_{n^{\prime}_{1}}$ such that
$[\sigma^{*}]\in{\rm Ker}_{n^{\prime}_{1}}.$ By a direct calculation using
Proposition 3.2.1 and the relations $\theta_{i}(\sigma)+\sigma\equiv 0$ for
$1\leq i\leq 4,$ we get

$[QP^{\otimes
4}_{n^{\prime}_{1}}]^{GL_{4}}=\langle[\widetilde{\varphi}(t_{1}t_{2}t_{3}^{6}t_{4}^{6}+t_{1}^{3}t_{2}^{3}t_{3}^{4}t_{4}^{4})]\rangle.$

Now the theorem follows from the fact that
$\langle[\widetilde{\varphi}(t_{1}t_{2}t_{3}^{6}t_{4}^{6}+t_{1}^{3}t_{2}^{3}t_{3}^{4}t_{4}^{4})],[\overline{\zeta}]\rangle=1.$
∎

###### Proof of Theorem 2.2.3.

The proof of this theorem is similar to the one in Theorem 2.1.4. Note that
the invariant space $[QP^{\otimes 4}_{n^{\prime}_{2}}]^{GL_{4}}$ generated
over $\mathbb{Z}/2$ by the class

$\big{[}\varphi\big{(}\sum_{1\leq\ell\leq 3}\sum_{1\leq i_{1}\leq\ldots\leq
i_{\ell}\leq 4}t_{i_{1}}t_{i_{2}}^{2}\ldots
t^{2^{\ell-2}}_{i_{\ell-1}}t^{32-2^{\ell-1}}_{i_{\ell}}+t_{1}t_{2}^{2}t_{3}^{4}t_{4}^{24}\big{)}\big{]}.$

∎

###### Proof of Theorem 2.2.5.

Firstly, we recall a result in our work [38] which is used in the proof of the
theorem.

######  Theorem 3.2.3.

Let $s$ and $t$ be positive integers such that $t\geq 4.$ Then,

$\dim\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{2^{s+t+1}+2^{s+1}-3}^{\otimes
4}]^{*}=\left\\{\begin{array}[]{ll}1&\mbox{if $s=1,2$},\\\ 2&\mbox{if $s\geq
3$}.\end{array}\right.$

Moreover, $\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt&\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{2^{s+t+1}+2^{s+1}-3}^{\otimes 4}]^{*}\\\
&=\left\\{\begin{array}[]{ll}\langle[x_{1}^{(0)}x_{2}^{(2^{s+1}-1)}x_{3}^{(2^{s+t}-1)}x_{4}^{(2^{s+t}-1)}]\rangle&\mbox{if
$s=1,2$},\\\\[2.84526pt]
\langle[x_{1}^{(0)}x_{2}^{(2^{s+1}-1)}x_{3}^{(2^{s+t}-1)}x_{4}^{(2^{s+t}-1)}],[x_{1}^{(0)}x_{2}^{(2^{s}-1)}x_{3}^{(2^{s}-1)}x_{4}^{(2^{s+t+1}-1)}]\rangle&\mbox{if
$s\geq 3$}.\end{array}\right.\end{array}$

Theorem 3.2.3 implies that

$\dim\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{\frac{n^{\prime}_{s}-4}{2}}^{\otimes
4}]^{*}=\left\\{\begin{array}[]{ll}1&\mbox{if $s=3,4$}\\\ 2&\mbox{if $s\geq
5$}.\end{array}\right.$

Then, by the inequality (7) and Proposition 3.2.1, we get

(9) $\dim\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n^{\prime}_{s}}^{\otimes
4}]^{*}\leq\left\\{\begin{array}[]{ll}1&\mbox{if $s=4$}\\\ 2&\mbox{if $s\neq
4$}.\end{array}\right.$

On the other hand, it is not too difficult to check that the elements

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt\zeta_{1}&=x_{1}^{(1)}x_{2}^{(7)}x_{3}^{(63)}x_{4}^{(63)}\in[P_{n^{\prime}_{3}}^{\otimes
4}]^{*},\ \
\zeta_{2}=x_{1}^{(0)}x_{2}^{(0)}x_{3}^{(7)}x_{4}^{(127)}\in[P_{n^{\prime}_{3}}^{\otimes
4}]^{*},\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\zeta_{3}&=x_{1}^{(1)}x_{2}^{(15)}x_{3}^{(127)}x_{4}^{(127)}\in[P_{n^{\prime}_{4}}^{\otimes
4}]^{*}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\zeta_{s}&=x^{(1)}x_{2}^{(2^{s-1}-1)}x_{3}^{(2^{s-1}-1)}x_{4}^{(2^{s+4}-1)}\in[P_{n^{\prime}_{s}}^{\otimes
4}]^{*},\ \mbox{for $s\geq 5$}.\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\zeta^{\prime}_{s}&=x_{1}^{(1)}x_{2}^{(2^{s}-1)}x_{3}^{(2^{s+3}-1)}x_{4}^{(2^{s+3}-1}\in[P_{n^{\prime}_{s}}^{\otimes
4}]^{*}\ \mbox{for $s\geq 5$}.\end{array}$

are $\overline{\mathcal{A}}$-annahilated. Then, using the representation in
$\Lambda$ of the rank 4 algebraic transfer, we get

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt&[\psi_{4}(\zeta_{1})]=[\lambda_{1}\lambda_{7}\lambda_{63}^{2}]=Tr_{4}^{\mathcal{A}}([\zeta_{1}])=h_{1}h_{3}h_{6}^{2}\in{\rm
Ext}_{\mathcal{A}}^{4,4+n^{\prime}_{3}}(\mathbb{Z}/2,\mathbb{Z}/2),\\\ \vskip
6.0pt plus 2.0pt minus
2.0pt&[\psi_{4}(\zeta_{2})]=[\lambda_{0}\lambda_{7}\lambda_{127}^{2}]=Tr_{4}^{\mathcal{A}}([\zeta_{2}])=h_{0}^{2}h_{3}h_{7}\in{\rm
Ext}_{\mathcal{A}}^{4,4+n^{\prime}_{3}}(\mathbb{Z}/2,\mathbb{Z}/2),\\\ \vskip
6.0pt plus 2.0pt minus
2.0pt&[\psi_{4}(\zeta_{3})]=[\lambda_{1}\lambda_{15}\lambda_{127}^{2}]=Tr_{4}^{\mathcal{A}}([\zeta_{3}])=h_{1}h_{4}h_{7}^{3}\in{\rm
Ext}_{\mathcal{A}}^{4,4+n^{\prime}_{4}}(\mathbb{Z}/2,\mathbb{Z}/2),\\\ \vskip
6.0pt plus 2.0pt minus
2.0pt&[\psi_{4}(\zeta_{s})]=[\lambda_{1}\lambda^{2}_{2^{s-1}-1}\lambda_{2^{s+4}-1}]=Tr_{4}^{\mathcal{A}}([\zeta_{s}])=h_{1}h_{s-1}^{2}h_{s+4}\in{\rm
Ext}_{\mathcal{A}}^{4,4+n^{\prime}_{s}}(\mathbb{Z}/2,\mathbb{Z}/2)\ \mbox{for
$s\geq 5$}\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt&[\psi_{4}(\zeta^{\prime}_{s})]=[\lambda_{1}\lambda_{2^{s}-1}\lambda_{2^{s+3}-1}^{2}]=Tr_{4}^{\mathcal{A}}([\zeta^{\prime}_{s}])=h_{1}h_{s}h_{s+3}^{2}\in{\rm
Ext}_{\mathcal{A}}^{4,4+n^{\prime}_{s}}(\mathbb{Z}/2,\mathbb{Z}/2)\ \mbox{for
$s\geq 5$}.\end{array}$

Combining these computations with the equality (8) and the inequality (9)
gives

$\dim\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n^{\prime}_{s}}^{\otimes
4}]^{*}=\left\\{\begin{array}[]{ll}1&\mbox{if $s=4$}\\\ 2&\mbox{if $s\neq
4$}.\end{array}\right.$

Furthermore, by using Proposition 3.2.1, Theorem 3.2.3 and the similar
arguments as in the proof of Theorem 2.2.1, we obtain

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus
2.0pt&\mathbb{Z}/2\otimes_{GL_{4}}{\rm
Ann}_{\overline{\mathcal{A}}}[P_{n^{\prime}_{s}}^{\otimes 4}]^{*}\\\
&=\left\\{\begin{array}[]{ll}\langle[x_{1}^{(1)}x_{2}^{(7)}x_{3}^{(63)}x_{4}^{(63)}],[x_{1}^{(0)}x_{2}^{(0)}x_{3}^{(7)}x_{4}^{(127)}]\rangle&\mbox{if
$s=3$},\\\\[2.84526pt]
\langle[x_{1}^{(1)}x_{2}^{(15)}x_{3}^{(127)}x_{4}^{(127)}]\rangle&\mbox{if
$s=4$},\\\\[2.84526pt]
\langle[x_{1}^{(1)}x_{2}^{(2^{s-1}-1)}x_{3}^{(2^{s-1}-1)}x_{4}^{(2^{s+4}-1)}],[x_{1}^{(1)}x_{2}^{(2^{s}-1)}x_{3}^{(2^{s+3}-1)}x_{4}^{(2^{s+3}-1)}]\rangle&\mbox{if
$s\geq 5$}.\end{array}\right.\end{array}$

The theorem is proved. ∎

##  4 Appendix

Let us recall that the structure of the cohomology ${\rm
Ext}_{\mathcal{A}}^{*,*}(\mathbb{Z}/2,\mathbb{Z}/2)$ of the Steenrod ring is
still a great mystery for all homological degrees $\geq 6.$ So far, Bruner’s
computer calculations [7] tell us only a little about the groups ${\rm
Ext}_{\mathcal{A}}^{h,*}(\mathbb{Z}/2,\mathbb{Z}/2)$ for $h\geq 6.$ The Singer
algebraic transfer $Tr_{h}^{\mathcal{A}}$ serves as a reliable tool for
depicting those Ext groups. This is concretized by the works of Singer [41],
and Boardman [4] that $Tr_{h}^{\mathcal{A}}$ is an isomorphism for all
homological degrees $h\leq 3$ and any internal degree $>0$. Singer [41]
predicted that the algebraic transfer is a monomorphism in every $h>0,$ but it
remains open for all $h\geq 5.$ Very little information is known this
conjecture for the rank 5 case (see also the present author [33, 34, 36, 39],
Sum [47]). In particular, it is no information yet on the cases $h=6,7,8.$ So,
in this appendix, we would like to investigate the behavior of the Singer
transfer of ranks $\geq 5$ in some internal degrees. More precisely, we first
study $Tr_{5}^{\mathcal{A}}$ in degrees of the form
$n_{s}:=k(2^{s}-1)+r.2^{s}$ with $k=4,\,5.$ Next, based on the known results
by Mothebe, Kaelo, and Ramatebele [26] and direct calculations, we investigate
the behavior of the sixth transfer, $Tr_{6}^{\mathcal{A}}$ in internal degrees
$\leq 25.$ At the same time, using the results by Bruner [7], we will show
that $Tr_{6}^{\mathcal{A}}$ is not an epimorphism in degrees $n_{s}$ with
$k=6,\,r=10,$ and $s\leq 2.$ More explicitly, $Tr_{6}^{\mathcal{A}}$ does not
detect the non-zero elements $h_{2}^{2}g_{1}=h_{4}Ph_{2}\in{\rm
Ext}_{\mathcal{A}}^{6,6+n_{1}}(\mathbb{Z}/2,\mathbb{Z}/2)$, and $D_{2}\in{\rm
Ext}_{\mathcal{A}}^{6,6+n_{2}}(\mathbb{Z}/2,\mathbb{Z}/2).$ Finally, we probe
the behavior of $Tr_{7}^{\mathcal{A}},$ and $Tr_{8}^{\mathcal{A}}$ in internal
degrees $\leq 15.$ As a consequence, Singer’s conjecture for the ranks
$h=6,7,8$ is true in respective degrees. Remarkably, based on the techniques
in this paper, we showed that the non-zero elements $h_{1}Ph_{1}\in{\rm
Ext}_{\mathcal{A}}^{6,6+10}(\mathbb{Z}/2,\mathbb{Z}/2)$, $h_{0}Ph_{2}\in{\rm
Ext}_{\mathcal{A}}^{6,6+11}(\mathbb{Z}/2,\mathbb{Z}/2)$ are not in the image
of $Tr_{6}^{\mathcal{A}},$ and that $h_{0}^{2}Ph_{2}\in{\rm
Ext}_{\mathcal{A}}^{6,6+11}(\mathbb{Z}/2,\mathbb{Z}/2)$ is not in the image of
$Tr_{7}^{\mathcal{A}}.$ These events have also been proved by Chơn, and Hà [9,
10] using another method, but their approaches were not directly applicable in
confirming or refuting the Singer conjecture in those cases. Therefore, our
calculations are important and of non-trivial value. Additionally, in ranks
$h=6,\,7,\,8,$ we prove that the results of Moetele, and Mothebe [27] for the
dimensions of the cohit $\mathbb{Z}/2$-module $QP_{13}^{\otimes h}$ are not
true. Therefrom, as described above, the behavior of the Singer transfer in
bidegree $(h,h+13)$ is obtained. Our results will play a crucial role in the
study of the Singer algebraic transfer in the certain generic internal
degrees. The calculation techniques are similar to the proofs of the main
results of this article and to our previous works [34, 36, 37, 39]. We
emphasize that along this section, because the dimensions of the cohit spaces
$QP^{\otimes h}$ in degrees $n$ given are too large, we do not list the bases
of these spaces in detail; more precisely, they are the equivalence classes of
admissible monomials of degrees $n$ in $\mathcal{A}$-modules $P^{\otimes h}$
for $5\leq h\leq 8.$ Moreover, the MAGMA computer algebra system [22] has also
been used for verifying the results.

For the reader’s convenience, some related knowledge are mentioned here, since
they will be needed later. We also refer the readers to the literatures [18],
[34] for more details. First of all, let us recall that the set of all the
admissible monomials of degree $n$ in $P^{\otimes h}$ is a minimal set of
$\mathcal{A}$-generators for $P^{\otimes h}$ in degree $n$. Therefore, the
space of indecomposables $QP^{\otimes h}_{n}$ has a basis consisting of all
the classes represent by the admissible monomials of degree $n$ in $P^{\otimes
h}.$ We are now going to review the following well-known. For a natural number
$k,$ writing $\alpha_{j}(k)$ and $\alpha(k)$ for the $j$-th coefficients and
the number of $1$’s in dyadic expansion of $k$, respectively. Thence,
$\alpha(k)=\sum_{j\geq 0}\alpha_{j}(k),$ and $k$ can be written in the form
$\sum_{j\geq 0}\alpha_{j}(k)2^{j},$ where $\alpha_{j}(k)$ belongs to
$\\{0,1\\},$ for all $j\geq 0.$ For a monomial
$t=t_{1}^{a_{1}}t_{2}^{a_{2}}\ldots t_{h}^{a_{h}}$ in $P_{n}^{\otimes h},$ we
consider a sequence associated with $t$ by
$\omega(t):=(\omega_{1}(t),\omega_{2}(t),\ldots,\omega_{i}(t),\ldots)$ where
$\omega_{i}(t)=\sum_{1\leq j\leq h}\alpha_{i-1}(a_{j})\leq h,$ for all $i\geq
1.$ This sequence is called the weight vector of $t.$ One defines
$\deg(\omega(t))=\sum_{j\geq 1}2^{j-1}\omega_{j}(t).$ We say that
$a(t)=(a_{1},\ldots,a_{h})$ is the exponent vector of $t.$ The sets of all
weight vectors and exponent vectors are given the left lexicographical order.
Assume that $t=t_{1}^{a_{1}}t_{2}^{a_{2}}\ldots t_{h}^{a_{h}}$ and
$t^{\prime}=t_{1}^{a^{\prime}_{1}}t_{2}^{a^{\prime}_{2}}\ldots
t_{h}^{a^{\prime}_{h}}$ are the monomials in $P_{n}^{\otimes h}.$ We say that
$t<t^{\prime}$ if and only if either $\omega(t)<\omega(t^{\prime})$ or
$\omega(t)=\omega(t^{\prime}),$ $a(t)<a^{\prime}(t^{\prime}).$ For a weight
vector $\omega$ of degree $n,$ we denote two subspaces associated with
$\omega$ by

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus 2.0ptP_{n}^{\otimes
h}(\omega)&=\langle\\{t\in P^{\otimes h}_{n}|\,\deg(t)=\deg(\omega)=n,\
\omega(t)\leq\omega\\}\rangle,\\\ P_{n}^{\otimes h}(<\omega)&=\langle\\{t\in
P^{\otimes h}_{n}|\,\deg(t)=\deg(\omega)=n,\
\omega(t)<\omega\\}\rangle.\end{array}$

Let $u$ and $v$ be two homogeneous polynomials in $P_{n}^{\otimes h}(\omega).$
We define the equivalence relation ”$\equiv_{\omega}$” on $P_{n}^{\otimes h}$
as $u\equiv_{\omega}v$ if and only if $(u+v)$ belongs to
$((\overline{\mathcal{A}}P_{n}^{\otimes h}\cap P_{n}^{\otimes
h}(\omega)+P_{n}^{\otimes h}(<\omega)).$ By this, one has a quotient space
$QP_{n}^{\otimes h}(\omega):=P_{n}^{\otimes
h}(\omega)/((\overline{\mathcal{A}}P_{n}^{\otimes h}\cap P_{n}^{\otimes
h}(\omega))+P_{n}^{\otimes h}(<\omega)),$ and an isomorphism $QP_{n}^{\otimes
h}\cong\bigoplus_{\deg(\omega)=n}QP_{n}^{\otimes h}(\omega).$ Due to
Definition 3.1 (see section two), we say that a monomial $t\in P_{n}^{\otimes
h}$ is inadmissible if there exist monomials $z_{1},z_{2},\ldots,z_{k}\in
P_{n}^{\otimes h}$ such that $z_{j}<t$ for $1\leq j\leq k$ and
$t\equiv\sum_{1\leq j\leq k}z_{j},$ from which $t$ is said to be admissible if
it is not inadmissible. Our motivating tool is the following.
$\bullet$ Kameko’s criterion [18] on inadmissible monomials: Let $t$ and $z$
be monomials in $P_{n}^{\otimes h}.$ For an integer $r>0,$ assume that there
exists an index $i>r$ such that $\omega_{i}(t)=0$, from which if $z$ is
inadmissible, then $tz^{2^{r}}$ is, too.

A monomial $z=\prod_{1\leq j\leq h}t_{j}^{\alpha_{j}}$ in $P^{\otimes h}_{n}$
is called a spike if every exponent $\alpha_{j}$ is of the form
$2^{\beta_{j}}-1.$ In particular, if the exponents $\beta_{j}$ can be arranged
to satisfy $\beta_{1}>\beta_{2}>\ldots>\beta_{r-1}\geq\beta_{r}\geq 1,$ where
only the last two smallest exponents can be equal, and $\beta_{j}=0$ for
$r<j\leq t,$ then $z$ is called a minimal spike. Another important tool to
recall here is that of relations on hit polynomials.
$\bullet$ Singer’s criterion [42] on hit monomials: Suppose that $t\in
P^{\otimes h}_{n}$ and $\mu(n)\leq h.$ Then, if $z$ is a minimal spike in
$P^{\otimes h}_{n}$ such that $\omega(t)<\omega(z),$ then $t\equiv 0$ (or in
other words, $t$ is hit).

It will be useful to adopt the following notations:

$\begin{array}[]{ll}\vskip 6.0pt plus 2.0pt minus 2.0pt(QP_{n}^{\otimes
h})^{0}:=\langle\\{[\prod_{1\leq j\leq h}t_{j}^{\alpha_{j}}]\in
QP_{n}^{\otimes h}|\,\deg(\prod_{1\leq j\leq
h}t_{j}^{\alpha_{j}})=n,\,\prod_{1\leq j\leq h}\alpha_{j}=0\\}\rangle,\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt(QP_{n}^{\otimes
h})^{>0}:=\langle\\{[\prod_{1\leq j\leq h}t_{j}^{\alpha_{j}}]\in
QP_{n}^{\otimes h}|\,\deg(\prod_{1\leq j\leq
h}t_{j}^{\alpha_{j}})=n,\,\prod_{1\leq j\leq h}\alpha_{j}\neq 0\\}\rangle,\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt(QP_{n}^{\otimes
h})^{0}(\omega):=\langle\\{[\prod_{1\leq j\leq h}t_{j}^{\alpha_{j}}]\in
QP_{n}^{\otimes h}(\omega)|\,\deg(\prod_{1\leq j\leq
h}t_{j}^{\alpha_{j}})=n,\,\prod_{1\leq j\leq
h}\alpha_{j}=0,\,\omega(\prod_{1\leq j\leq
h}t_{j}^{\alpha_{j}})=\omega\\}\rangle,\\\ (QP_{n}^{\otimes
h})^{>0}(\omega):=\langle\\{[\prod_{1\leq j\leq h}t_{j}^{\alpha_{j}}]\in
QP_{n}^{\otimes h}(\omega)|\,\deg(\prod_{1\leq j\leq
h}t_{j}^{\alpha_{j}})=n,\,\prod_{1\leq j\leq h}\alpha_{j}\neq
0,\,\omega(\prod_{1\leq j\leq
h}t_{j}^{\alpha_{j}})=\omega\\}\rangle,\end{array}$

where $\omega$ is a weight vector of degree $n.$ Then, we have the
isomorphisms:

$QP_{n}^{\otimes h}\cong(QP_{n}^{\otimes h})^{0}\,\bigoplus\,(QP_{n}^{\otimes
h})^{>0},\ \mbox{ and }\ QP_{n}^{\otimes h}(\omega)\cong(QP_{n}^{\otimes
h})^{0}(\omega)\,\bigoplus\,(QP_{n}^{\otimes h})^{>0}(\omega).$

Denote the set of all the admissible monomials of degree $n$ in
$\mathcal{A}$-module $P^{\otimes h}$ by $\mathscr{C}_{n}^{\otimes h}.$ For a
monomial $t\in P^{\otimes h}_{n},$ we denote by $[t]$ the equivalence class of
$t$ in $QP_{n}^{\otimes h}.$ If $\omega$ is a weight vector of degree $n$ and
$\omega(t)=\omega,$ then denote by $[t]_{\omega}$ the equivalence class of $t$
in $QP_{n}^{\otimes h}(\omega).$ As usual, we write $|\mathscr{C}_{n}^{\otimes
h}|$ for the cardinality of $\mathscr{C}_{n}^{\otimes h},$ and put
$[\mathscr{C}_{n}^{\otimes h}]=\\{[t]\,:\,t\in\mathscr{C}_{n}^{\otimes h}\\}.$
For any admissible monomials ${\rm adm}_{1},\ldots,{\rm adm}_{k}$ in
$P^{\otimes h}_{n}(\omega)$ and for a subgroup $\mathscr{G}$ of $GL_{h}.$ Let
us denote by $\mathscr{G}({\rm adm}_{1},\ldots,{\rm adm}_{k})$ the
$\mathbb{Z}/2\mathscr{G}$-submodule of $QP_{n}^{\otimes h}(\omega)$ generated
by $\\{[{\rm adm}_{j}]_{\omega}:\,1\leq j\leq k\\}.$ It should be noted that
if $\omega$ is a weight vector of a minimal spike, then $[{\rm
adm}_{j}]_{\omega}=[{\rm adm}_{j}]$ for all $j.$

###  4.1 The behavior of $Tr_{5}^{\mathcal{A}}$ in some internal degrees

The goal of this section is to study the fifth Singer transfer in some
internal degrees of the form $m_{r,s}:=5(2^{s}-1)+r.2^{s}$ with $s=0,$ and
$r\in\\{30,\,33,\,34,\,36,\,38,\,39,\,40,\,42,\,45,46,\,47,\,48\\}.$ A direct
computation using previous results by the present author [34, 35, 36],
Mothebe, and Uys [25], Sum [43, 44, 45, 47], Ly, and Tin [21], Tin [48, 49],
we find that

######  Theorem 4.1.1.