Normalized Derivations for Steenrod--Milnor Operations on the Dickson Algebra and Applications

Abstract

Computing higher iterates of the primitive Steenrod-Milnor operations $\mathrm{St}^{\emptyset,\Delta_i}$ on the Dickson algebra $D_n$ has historically been obstructed by severe combinatorial complexity. We resolve this for all $i \ge 1$ by introducing a normalized derivation $\delta_i = (-1)^n Q_{n,0}^{-1} \mathrm{St}^{\emptyset,\Delta_i}$. Starting from a recent explicit formula of Nguyen Sum, we show that $\delta_i$ preserves $D_n$ and transforms the non-linear action into a tractable affine differential system. Resolving the non-commutativity of the operators via unsigned Stirling numbers of the first kind, we derive closed formulas for all higher iterates, systematically yielding the ambient nilpotence identity $(\mathrm{St}^{\emptyset,\Delta_i})^p = 0$ on $D_n$. Furthermore, $\delta_i$ satisfies a restricted $p$-power relation $\delta_i^p = B^{p-1}\delta_i$, where $B = R_{n,i}^p$. When $B \neq 0$, localizing at $B$ yields an explicit $\mathbb{F}_p$-weight decomposition of the localized Dickson algebra via Lagrange projectors, providing exact kernel and image descriptions for both operators. As applications, we explicitly determine the kernel and image of $\delta_i$ in the classical range $1 \le i \le n$. This formalism unifies and upgrades the classical first-order formulas of Smith-Switzer and Wilkerson, along with Sum's low extra formulas ($i=n+1, n+2$), into closed expressions for all iterates. Finally, we construct a commutative-algebra Koszul complex from the normalized-ratio coefficients, clarifying its structural distinction from the square-zero Margolis complexes recently studied in the literature.