Structure of the space of $G{L}_4({\mathbb{Z}}_2)$-coinvariants ${\mathbb{Z}}_2{\otimes }_{G{L}_4({\mathbb{Z}}_2)}P{H}_{\ast }({\mathbb{Z}}_2^4,{\mathbb{Z}}_2)$ in some generic degrees and its application

Let $A$ denote the Steenrod algebra at the prime 2 and let $k={\mathbb{Z}}_2\mathrm{.}$ An open problem of homotopy theory is to determine a minimal set of $A$-generators for the polynomial ring $P_q = k[x_1, \ldots, x_q] = H^{}(k^{q}, k)$on$q$generators$x_1, \ldots, x_q$with$|x_i|= 1.$Equivalently,onecanwritedownexplicitlyabasisforthegradedvectorspace$Q^{\otimes q} := k\otimes_{A} P_q$ineachnon-negativedegree$n.$ This is the content of the classical “hit problem” in literature [30]. Based on this problem, we are interested in the $q$-th cohomological transfer $T{r}_q^A$ of Singer [39], which is one of the useful tools for describing mod-2 cohomology of the algebra $A\mathrm{.}$ This transfer is a linear map from the space of $G{L}_q(k)$-coinvariant $k{\otimes }_{G{L}_q(k)}P((P_q{)}_n^{\ast })$ of $Q^{\otimes q}$ to the $k$-cohomology group of the Steenrod algebra, ${\rm Ext}_{A}^{q, q+n}(k, k).$Here$GL_q(k)$isthegenerallineargroupofdegree$q$overthefield$k,$and$P((P_q)_n^{})$istheprimitivepartof$(P_q)^{*}_n$ under the action of $A\mathrm{.}$ Singer conjectured that $T{r}_q^A$ is a monomorphism, but this remains unanswered for all $q\ge 4.$ The present paper is to devoted to the investigation of this conjecture for the rank 4 case. More specifically, basing the techniques of the hit problem of four variables, we explicitly determine the structure of $k{\otimes }_{G{L}_4(k)}P((P_4{)}_n^{\ast })$ in some generic degrees $n\mathrm{.}$ Applying these results and a representation of $T{r}_4^A$ over the lambda algebra, we notice that Singer’s conjecture is true for the rank 4 transfer in those degrees $n$. Also, we give some conjectures on the dimensions of $k{\otimes }_{G{L}_q(k)}((P_4{)}_n^{\ast })$ for the remaining degrees $n\mathrm{.}$ As a consequence, Singer’s conjecture holds for $T{r}_4^A\mathrm{.}$ This study and our previous results have been provided a panorama of the behavior of the fourth cohomological transfer.