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

Let $\mathcal{A}$ denote the classical singly-graded Steenrod algebra over the binary field $\mathbb{Z}\mathrm{/}2.$ We write $P_k:=\mathbb{Z}\mathrm{/}2[t_1,t_2,\dots ,t_k]$ as the polynomial algebra on $k$ generators, each having a degree of one. Let $G{L}_k$ be the general linear group of rank $k$ over $\mathbb{Z}\mathrm{/}2.$ Then, $P_k$ is an $\mathcal{A}[G{L}_k]$-module. The structure of the cohomology groups, ${\mathrm{E}\mathrm{x}\mathrm{t}}_{\mathcal{A}}^{k,k+\bullet }(\mathbb{Z}\mathrm{/}2,\mathbb{Z}\mathrm{/}2)$, 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: $$T{r}_k:\mathrm{H}\mathrm{o}\mathrm{m}([(\mathbb{Z}\mathrm{/}2{\otimes }_{\mathcal{A}}{P}_k{)}_{\bullet }{]}^{G{L}_k},\mathbb{Z}\mathrm{/}2)⟶{\mathrm{E}\mathrm{x}\mathrm{t}}_{\mathcal{A}}^{k,k+\bullet }(\mathbb{Z}\mathrm{/}2,\mathbb{Z}\mathrm{/}2)\mathrm{.}$$ Among Singer’s contributions is an interesting open conjecture asserting the monomorphism of $T{r}_k$ for all $k\mathrm{.}$ 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)].