Normalized Derivations for Milnor's Primitive Operations on the Dickson Algebra and Applications
Dang Vo Phuc
Abstract
We study the action of the primitive Steenrod--Milnor operation $\mathrm{St}^{Δ_i}$ on the Dickson algebra $D_n=\mathbb{F}_p[Q_{n,0},Q_{n,1},\dots,Q_{n,n-1}].$ Starting from Sum's explicit formula on the Dickson generators, we observe that after dividing by $Q_{n,0}$ one obtains a genuine derivation on the localization $D_n[Q_{n,0}^{-1}].$ This normalized derivation provides a transparent framework for the action of $\mathrm{St}^{Δ_i}.$ Using this viewpoint, we derive a closed formula for all higher iterates of $\mathrm{St}^{Δ_i}$ on the Dickson generators, with an explicit factorial term and the resulting vanishing $(\mathrm{St}^{Δ_i})^m=0$ for all $m\ge p.$ We also obtain general kernel and image constructions, as well as a family of normalized ratios on which the action becomes affine-linear. When $B=R_{n,i}^{\,p}$ is invertible, we show that the normalized action is of Euler type on the family $Q_{n,0}^mΦ(I_1,\dots,I_{n-1}),$ where the $I_s$ are invariant ratios. In the classical range $2\le i<n$, this yields an explicit description of the kernel and image of the normalized derivation, while for $i=n$ it gives a grading-theoretic description. As an application, we show that this formalism recovers and strengthens several formulas of Sum: in the ranges $2\le i\le n$ and $i=n+1,n+2$, the known first-order identities extend to closed formulas for all higher iterates. We also explain a precise Koszul-type analogy with recent work of Ngo Anh Tuan. Unlike the Milnor primitives $Q_j$ arising in Margolis homology, the operation $\mathrm{St}^{Δ_i}$ need not square to zero, so the final section is formulated as a formal Koszul construction rather than a direct Margolis-homology computation.