A note on the hit problem for the polynomial algebra of six variables and the sixth algebraic transfer

Let A be the classical, singly-graded Steenrod algebra over prime order field F2 and let P⊗h:=F2[t1,…,th] denote polynomial on h generators, each of degree 1, viewed as a module A. Write GLh for usual general linear group rank F2. As well known, (mod 2) cohomology groups algebra, ExtAh,h+⁎(F2,F2) is still largely mysterious all homological degrees h≥6. The h-th algebraic transferTrhA:(F2⊗GLhAnnA‾[P⊗h]⁎)n⟶ExtAh,h+n(F2,F2), defined by William Singer [28], helpful tool to describe that Ext groups. conjectured this transfer monomorphism, but it remains open any h≥5. There currently no information conjecture h=6. In paper, we verify Singer's h=6 internal form ns:=6(2s−1)+10⋅2s. This result important, since tells us non-zero elements h22g1=h4Ph2∈ExtA6,6+n1(F2,F2), D2∈ExtA6,6+n2(F2,F2) are not in image sixth transfer. Note continuation our previous one [22], which will refer Part I. Our approach based explicitly solving hit problem case ns. extends Mothebe, Kaelo Ramatebele [13]. At same time, also use obtained establish dimension space unhit elements, F2⊗AP⊗7 certain degree.