Binary Minor Certificates for the 𝒜(1)-hit problem of MO(2)

Abstract

Let M=H̃^*(MO(2);𝔽₂) and let 𝒜(1) be the subalgebra of the mod $2$ Steenrod algebra generated by Sq¹ and Sq². In this work, we determine the quotient 𝔽₂⊗_𝒜(1)M in every internal degree h≥0, where internal degree h corresponds to cohomological degree h+2. To overcome the limitations of finite experimental extrapolation, our proof is strictly certifying: in each degree, we exhibit a determinant-one minor of the binary hit matrix and a complementary family of linear dual functionals annihilating every hit column, mathematically guaranteeing the exact rank for all degrees. This rigorously establishes the monomial generating families and module decomposition proposed in Shih's 2012 report. Furthermore, we derive a closed binomial formula for every Steenrod square from the length-two symmetric-polynomial model, correct a boundary omission in the 𝒜(0) calculation, compute the exact Hilbert series of the 𝒜(1)-cohit, and prove an intrinsic direct-sum decomposition of M into two stable 𝒜(1)-submodules.

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BibTeX RIS

BibTeX

@misc{phuc2026binary,
  title = {{Binary Minor Certificates for the $\mathcal A(1)$-hit problem of $MO(2)$}},
  author = {Dang Vo Phuc},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20859},
  eprint = {2609.20859},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20859},
}

RIS

TY  - UNPB
TI  - Binary Minor Certificates for the 𝒜(1)-hit problem of MO(2)
AU  - Dang Vo Phuc
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.20859
UR  - https://arxiv.org/abs/2609.20859
AB  - Let M=H̃^*(MO(2);𝔽_2) and let 𝒜(1) be the subalgebra of the mod $2$ Steenrod algebra generated by Sq^1 and Sq^2. In this work, we determine the quotient 𝔽_2⊗_𝒜(1)M in every internal degree h≥0, where internal degree h corresponds to cohomological degree h+2. To overcome the limitations of finite experimental extrapolation, our proof is strictly certifying: in each degree, we exhibit a determinant-one minor of the binary hit matrix and a complementary family of linear dual functionals annihilating every hit column, mathematically guaranteeing the exact rank for all degrees. This rigorously establishes the monomial generating families and module decomposition proposed in Shih's 2012 report. Furthermore, we derive a closed binomial formula for every Steenrod square from the length-two symmetric-polynomial model, correct a boundary omission in the 𝒜(0) calculation, compute the exact Hilbert series of the 𝒜(1)-cohit, and prove an intrinsic direct-sum decomposition of M into two stable 𝒜(1)-submodules.
ER  -

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