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Higher Labute-Serre duality and Lyndon words

  • Ido Efrat
  • Levav Ferber Tas
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Abstract

Let S be a free profinite group on a finite ordered basis X, and let S(n,p), n=1,2,…, denote its lower p-central filtration. There is a natural duality between S(n,p)/S(n+1,p) and H2(S/S(n,p),𝔽p). These 𝔽p-linear spaces admit natural bases indexed by Lyndon words of length ≤ n in the alphabet X. These bases are known to be unitriangularly dual. We prove that they are much closer to being fully dual, by showing that the pairing between two basis elements vanishes unless the corresponding Lyndon words are permutations of one another. We further show that the value of the pairing is essentially independent of n.

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Cite

BibTeXRIS
BibTeX
@misc{efrat2026higher,
  title = {{Higher Labute-Serre duality and Lyndon words}},
  author = {Ido Efrat and Levav Ferber Tas},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.21695},
  eprint = {2609.21695},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21695},
}
RIS
TY  - UNPB
TI  - Higher Labute-Serre duality and Lyndon words
AU  - Ido Efrat
AU  - Levav Ferber Tas
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.21695
UR  - https://arxiv.org/abs/2609.21695
AB  - Let S be a free profinite group on a finite ordered basis X, and let S^(n,p), n=1,2,…, denote its lower p-central filtration. There is a natural duality between S^(n,p)/S^(n+1,p) and H^2(S/S^(n,p),𝔽_p). These 𝔽_p-linear spaces admit natural bases indexed by Lyndon words of length ≤ n in the alphabet X. These bases are known to be unitriangularly dual. We prove that they are much closer to being fully dual, by showing that the pairing between two basis elements vanishes unless the corresponding Lyndon words are permutations of one another. We further show that the value of the pairing is essentially independent of n.
ER  -