Article
| Title: | A note on harmonic patterns and multi-variable formulae for the action of Steenrod powers (English) |
| Author: | Ðăng, Phúc Vő |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 76 |
| Issue: | 2 |
| Year: | 2026 |
| Pages: | 645-661 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| 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 $p=3$. (English) |
| Keyword: | Steenrod algebra |
| Keyword: | Steenrod power |
| Keyword: | divided power algebra |
| MSC: | 55S05 |
| MSC: | 55S10 |
| DOI: | 10.21136/CMJ.2026.0391-25 |
| . | |
| Date available: | 2026-05-22T11:24:19Z |
| Last updated: | 2026-05-25 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153654 |
| . | |
| Reference: | [1] Azizi, S., Janfada, A. S.: Up and down Steenrod powers.J. Math. Ext. 18 (2024), Article ID 1, 19 pages. Zbl 08047540, 10.30495/JME.2024.3206 |
| Reference: | [2] Karaca, I.: Monomial bases in the mod-$p$ Steenrod algebra.Czech. Math. J. 55 (2005), 699-707. Zbl 1081.55016, MR 2153094, 10.1007/s10587-005-0057-2 |
| Reference: | [3] Karaca, I., Karaca, I. Y.: On conjugation in the mod-$p$ Steenrod algebra.Turk. J. Math. 24 (2000), 359-365. Zbl 0971.55019, MR 1803818 |
| Reference: | [4] Oner, T., Tanay, B.: $P$-matrices for the action of Steenrod power operations on polynomial algebra.J. Math. Syst. Sci. 3 (2013), 543-549. |
| Reference: | [5] Palmieri, J. H., Zhang, J. J.: Commutators in the Steenrod algebra.New York J. Math. 19 (2013), 23-37. Zbl 1284.55020, MR 3028133 |
| Reference: | [6] Turgay, N. D.: On the mod $p$ Steenrod algebra and the Leibniz-Hopf algebra.Electron. Res. Arch. 28 (2020), 951-959. Zbl 1453.55013, MR 4128404, 10.3934/era.2020050 |
| Reference: | [7] Turgay, N. D., Karaca, I.: The Arnon bases in the Steenrod algebra.Georgian Math. J. 27 (2018), 649-654. Zbl 1476.55041, MR 4168725, 10.1515/gmj-2018-0076 |
| Reference: | [8] Walker, G., Wood, R. M. W.: The nilpotence height of $P^{p^{n}}$.Math. Proc. Camb. Philos. Soc. 123 (1998), 85-93. Zbl 0892.55008, MR 1474867, 10.1017/S0305004197001813 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
