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Keywords:
Steenrod algebra; Steenrod power; divided power algebra
Steenrod algebra; Steenrod power; divided power algebra
Summary:
This note extends the recent work of S. Azizi, A. S. Janfada (2024) on the symmetric treatment of ``up'' and ``down'' Steenrod powers for an odd prime $p$. We give a rigorous proof that their recursively defined Triangular algorithm agrees with the algebraic action of the up Steenrod powers $\mathcal {P}^k$ on polynomial algebras, thereby formalizing the harmonic patterns they observed. Building on this, we establish a multivariable extension of their one-variable formula: for a monomial $x^\alpha $ and any $k\ge 0$, the Cartan-Lucas factorization yields an explicit expansion of $\mathcal {P}^k(x^\alpha )$ whose nonvanishing is governed coordinatewise by the digitwise partial order $\preceq _p$. For a general polynomial $f$, we obtain a support-level description $$ {\rm supp}(\mathcal {P}^k(f))\subseteq \mathcal {S}_k(f) =\{\beta =\alpha +(p-1)\kappa \colon \alpha \in {\rm supp}(f), |\kappa |=k, \kappa _i\preceq _p\alpha _i \}, $$ together with an explicit coefficient formula. On the combinatorial side, we identify the $0/1$ triangular matrices $[U_p](t)$ with the Kronecker powers $T_p^{\otimes t}$ of the $p\times p$ upper-triangular all-ones matrix $T_p$, proving the digitwise characterization $[U_p](t)_{k,d}=1\iff k\preceq _p d$. Via graded duality, the same digitwise criterion yields an analogous support-level description for the down Steenrod powers $\mathcal {P}_k$ on the divided power algebra $DP(n)$, and we illustrate the resulting row-shift dictionary between up and down patterns by explicit $0/1$ heatmaps for $p=3$.
References:
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