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Let $A$ denote the Steenrod algebra at the prime 2 and let $k = \mathbb Z_2.$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, one can write down explicitly a basis for the graded vector space$Q^{\otimes q} := k\otimes_{A} P_q$ in each non-negative degree $n.$ This isthe content of the classical "hit problem" in literature [30]. Based on thisproblem, we are interested in the $q$-th cohomological transfer $Tr_q^{A}$ ofSinger [39], which is one of the useful tools for describing mod-2 cohomologyof the algebra $A.$ This transfer is a linear map from the space of$GL_q(k)$-coinvariant $k\otimes _{GL_q(k)} P((P_q)_n^{*})$ of $Q^{\otimes q}$to the $k$-cohomology group of the Steenrod algebra, ${\rm Ext}_{A}^{q, q+n}(k,k).$ Here $GL_q(k)$ is the general linear group of degree $q$ over the field$k,$ and $P((P_q)_n^{*})$ is the primitive part of $(P_q)^{*}_n$ under theaction of $A.$ Singer conjectured that $Tr_q^{A}$ is a monomorphism, but thisremains unanswered for all $q\geq 4.$ The present paper is to devoted to theinvestigation of this conjecture for the rank 4 case. More specifically, basingthe techniques of the hit problem of four variables, we explicitly determinethe structure of $k\otimes _{GL_4(k)} P((P_4)_{n}^{*})$ in some generic degrees$n.$ Applying these results and a representation of $Tr_4^{A}$ over the lambdaalgebra, we notice that Singer's conjecture is true for the rank 4 transfer inthose degrees $n$. Also, we give some conjectures on the dimensions of$k\otimes_{GL_q(k)} ((P_4)_n^{*})$ for the remaining degrees $n.$ As aconsequence, Singer's conjecture holds for $Tr_4^{A}.$ This study and ourprevious results have been provided a panorama of the behavior of the fourthcohomological transfer.

Structure of the space of $GL_4(\mathbb Z_2)$-coinvariants $\mathbb  Z_2\otimes_{GL_4(\mathbb Z_2)} PH_*(\mathbb Z_2^4, \mathbb Z_2)$ in some  generic degrees and its application