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Solomon--Terao Algebras: Top Degree, Dimension, and Homology

  • Hanyu Cai
  • Haiyang Chen
  • Shixuan Zeng
  • Shanwei Zhang
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Abstract

We study the Solomon--Terao algebra and the Solomon--Terao complex of a central hyperplane arrangement π’œ in β„‚β„“. Let Ξ· be a homogeneous polynomial of degree dβ‰₯2 whose restriction to every flat of π’œ has no critical point away from the origin. We prove that the Solomon--Terao algebra ST(π’œ,Ξ·) has top degree |π’œ|+β„“(dβˆ’2) and a one-dimensional top graded component with an explicit generator, solving a conjecture of Abe, Maeno, Murai, and Numata. For d=2, we show that the total dimension of ST(π’œ,Ξ·) is bounded below by that of the Orlik--Solomon algebra. Both results hold without assuming that the arrangement π’œ is tame. Finally, we prove that π’œ is tame if and only if all positive-degree homology groups of the Solomon--Terao complex associated with Ξ· vanish.

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Cite

BibTeXRIS
BibTeX
@misc{cai2026solomonterao,
  title = {{Solomon--Terao Algebras: Top Degree, Dimension, and Homology}},
  author = {Hanyu Cai and Haiyang Chen and Shixuan Zeng and Shanwei Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16951},
  eprint = {2610.16951},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16951},
}
RIS
TY  - UNPB
TI  - Solomon--Terao Algebras: Top Degree, Dimension, and Homology
AU  - Hanyu Cai
AU  - Haiyang Chen
AU  - Shixuan Zeng
AU  - Shanwei Zhang
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16951
UR  - https://arxiv.org/abs/2610.16951
AB  - We study the Solomon--Terao algebra and the Solomon--Terao complex of a central hyperplane arrangement π’œ in β„‚^β„“. Let Ξ· be a homogeneous polynomial of degree dβ‰₯2 whose restriction to every flat of π’œ has no critical point away from the origin. We prove that the Solomon--Terao algebra ST(π’œ,Ξ·) has top degree |π’œ|+β„“(dβˆ’2) and a one-dimensional top graded component with an explicit generator, solving a conjecture of Abe, Maeno, Murai, and Numata. For d=2, we show that the total dimension of ST(π’œ,Ξ·) is bounded below by that of the Orlik--Solomon algebra. Both results hold without assuming that the arrangement π’œ is tame. Finally, we prove that π’œ is tame if and only if all positive-degree homology groups of the Solomon--Terao complex associated with Ξ· vanish.
ER  -