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PreprintCC BY 4.0arXiv

Combinatorics of hyperplane arrangements and Witten zeta function at the origin

  • Kam Cheong Au
  • Kazuhiro Onodera
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Abstract

We introduce a new method that brings the combinatorics of hyperplane arrangements into the study of representation zeta functions of compact Lie groups. For the Witten zeta function ζΦ(s) associated with a root system Φ, our method yields elegant formulas for ζΦ(0) and ζΦ'(0) in terms of the exponents of various parabolic subsystems of Φ. Such formulas do not appear to be readily accessible through the conventional analytic techniques in the literature. More generally, the method applies to a broad family of conical zeta functions, expressing these two special values through the Möbius function of the intersection poset of the associated hyperplane arrangement.

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Cite

BibTeXRIS
BibTeX
@misc{au2026combinatorics,
  title = {{Combinatorics of hyperplane arrangements and Witten zeta function at the origin}},
  author = {Kam Cheong Au and Kazuhiro Onodera},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20740},
  eprint = {2609.20740},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20740},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Combinatorics of hyperplane arrangements and Witten zeta function at the origin
AU  - Kam Cheong Au
AU  - Kazuhiro Onodera
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.20740
UR  - https://arxiv.org/abs/2609.20740
AB  - We introduce a new method that brings the combinatorics of hyperplane arrangements into the study of representation zeta functions of compact Lie groups. For the Witten zeta function ζ_Φ(s) associated with a root system Φ, our method yields elegant formulas for ζ_Φ(0) and ζ_Φ'(0) in terms of the exponents of various parabolic subsystems of Φ. Such formulas do not appear to be readily accessible through the conventional analytic techniques in the literature. More generally, the method applies to a broad family of conical zeta functions, expressing these two special values through the Möbius function of the intersection poset of the associated hyperplane arrangement.
ER  -