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 -- constructed by W. Singer in [Math. Z. 202, 493--523 (1989)] -- plays an important role. The Singer transfer is represented 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 $k.$ For this reason, our main aim in this article is to ascertain the validity of the Singer conjecture for ranks 4 and 6 in certain families of internal degrees. We place particular emphasis on the rank 4 case. More precisely, we present a detailed proof for certain generic degree cases when verifying the conjecture of rank four, which were succinctly noted in our previous work [Proc. Roy. Soc. Edinburgh Sect. A 153, 1529--1542 (2023)].