Matrix methods for the up and down Steenrod squares

Let P(n) be the polynomial algebra which is a graded left module over the Steenrod algebra. The divided power algebra DPd(n) is defined as the Hopf dual of Pd(n). The dual of the Steenrod square Sqk is the linear map Sqk : DPd+k(n) -> DPd(n), called the down Steenrod square, defined by Sqk(u) = v for u is an element of DPd+k(n), where v(f) = (Sqk(u))(f) = u(Sqk(f )) for f is an element of Pd(n). In this article we consider the down Steenrod squares and establish some results about. Further, we show that there is some periodic calculation on the down Steenrod squares hanging on the limited leading period. Finally, using this fact, we exhibit a matrix method for manipulating the down Steenrod squares.