On the dimension of [formula omitted] as a module over Steenrod ring

Abstract

We write P for the polynomial algebra in one variable over the finite field Z2 and P⊗t=Z2[x1,…,xt] for its t-fold tensor product with itself. We grade P⊗t by assigning degree 1 to each generator. We are interested in determining a minimal set of generators for the ring of invariants (P⊗t)Gt as a module over Steenrod ring, A2. Here Gt is a subgroup of the general linear group GL(t,Z2). An equivalent problem is to find a monomial basis of the space of “unhit” elements, Z2⊗A2(P⊗t)Gt in each t and degree n≥0. The structure of this tensor product is proved surprisingly difficult and has been not yet known for t≥5, even for the trivial subgroup Gt={e}. In the present paper, we consider the subgroup Gt={e} for t∈{5,6}, and obtain some new results on A2-generators of (P⊗t)Gt in some degrees. At the same time, some of their applications have been proposed. We also provide an algorithm in MAGMA for verifying the results. This study can be understood as a continuation of our recent works in [24,26].