Up and down Steenrod powers

Abstract

First we review Steenrod powers, called in this article as the up Steenrod powers, and prove some more properties of them. Then, the divided $p$-power algebras are introduced and the down Steenrod powers, the dual of up Steenrod powers, are defined over these algebras. Finding some efficient tools for calculating the evaluations of up and down powers is the next attempt. Finally, harmonic patterns are exhibited for the action of up and down powers. All considerations are performed for one variable since Cartan formula supports naturally multi-variable circumstances.