On the algebraic transfers of ranks 4 and 6 at generic degrees

Let $\mathscr A$ denote the classical singly-graded Steenrod algebra over the binary field $\mathbb Z/2.$ We write $P_k:=\mathbb Z/2[t_1, t_2, \ldots, t_k]$ as the polynomial algebra on $k$ generators, each having a degree of one. Let $GL_k$ be the general linear group of rank $k$ over $\mathbb Z/2.$ Then, $P_k$ is an $ \mathscr A[GL_k]$-module. The structure of the cohomology groups, ${\rm Ext}_{ \mathscr A}^{k, k+\bullet}(\mathbb Z/2, \mathbb Z/2)$, of the Steenrod algebra has, thus far, resisted clear understanding and full description for all homological degrees $k$. In the study of these groups, the algebraic transfer\textemdash constructed by W. Singer in [Math. Z. \textbf{202}, 493--523 (1989)]\textemdash plays an important role. The Singer transfer is defined by the following homomorphism:$$Tr_k: {\rm Hom}([(\mathbb Z/2\otimes_{ \mathscr A} P_k)_{\bullet}]^{GL_k}, \mathbb Z/2)\longrightarrow {\rm Ext}_{ \mathscr A}^{k, k+\bullet}(\mathbb Z/2, \mathbb Z/2).$$ Among Singer's contributions is an interesting open conjecture asserting the monomorphism of $Tr_k$ for all $k.$ Motivated by this context, our main aim in this article is to prove the Singer conjecture for ranks 4 and 6 in certain families of internal degrees. In particular, for the rank 4 case, we discuss the indecomposable element $p_0\in \operatorname{Ext}_{\mathscr{A}}^{4,37}(\mathbb{Z}/2, \mathbb{Z}/2)$ in connection with the prior work of Hung and Quynh~\cite{Hung2}. Our rigorous computational proof that $p_0\in \operatorname{Im}(Tr_4)$ is new. For the rank 6 case, we show that $Tr_6$ detects elements in the $Sq^{0}$-family initiated by the class $h_5Ph_1\in \operatorname{Ext}_{\mathscr{A}}^{6,46}(\mathbb{Z}/2, \mathbb{Z}/2).$ All principal results presented herein have been verified through our recently developed algorithms implemented on the computer algebra system \texttt{OSCAR}. We also provide detailed \texttt{OSCAR} code and usage notes for readers in the Appendix.