On the hit problem for the Steenrod algebra in the generic degree and its applications

Let P n be the graded polynomial algebra over the prime field of two elements, F 2 . We investigate the Peterson hit problem for the polynomial algebra P n , viewed as a module over the mod-2 Steenrod algebra, A . This problem remains unsolvable for n > 4 , even with the aid of computers in the case of n = 5 . The goal of this work is to develop a result in Phuc (J Korean Math Soc 58(3):643–702, 2021) for P n in the generic degree d s = a ( 2 s - 1 ) + b 2 s + 1 , where a = n = 5 , b = 11 , and s is an arbitrary non-negative integer. This topic has been investigated by Phuc (2021) for s = 0 . We also use the aforementioned results to establish the dimension result for the vector space F 2 ⊗ A P n for the case n = 6 in some degrees. The hit problem is used to investigate Singer’s homomorphism, which is a homomorphism from the homology of the Steenrod algebra to the subspace of ( F 2 ⊗ A P n ) d consisting of all the G L ( n ; F 2 ) -invariant classes. It is useful for explaining the homology groups of the Steenrod algebra, Tor n , n + d A ( F 2 , F 2 ) . The behavior of the fifth Singer algebraic transfer in degree 5 ( 2 s - 1 ) + 11 · 2 s + 1 is also discussed at the conclusion of this study.