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

Abstract

Computing higher iterates of the primitive Steenrod-Milnor operations ${\mathrm{St}}^{\varnothing ,{\Delta }_i}$ 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 ${\delta }_i=(-1{)}^n{Q}_{n,0}^{-1}{\mathrm{St}}^{\varnothing ,{\Delta }_i}$. 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 $({\mathrm{St}}^{\varnothing ,{\Delta }_i}{)}^p=0$ 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 $B\ne 0$, 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.

arXiv comment30 pages. In this final version, further references and new structural and computational results have been incorporated. Constructive comments and discussions are always appreciated