{"ID":23646983,"CreatedAt":"2026-09-18T07:39:37.512926072Z","UpdatedAt":"2026-09-18T07:39:37.512926072Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.20740","arxiv_id":"2609.20740","title":"Combinatorics of hyperplane arrangements and Witten zeta function at the origin","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 $ζ_Φ(s)$ 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.","short_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 $ζ_Φ(s)$ 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 va...","url_abs":"https://arxiv.org/abs/2609.20740","url_pdf":"https://arxiv.org/pdf/2609.20740v1","authors":"[\"Kam Cheong Au\",\"Kazuhiro Onodera\"]","published":"2026-09-17T17:26:54Z","proceeding":"math.CO","tasks":"[\"math.CO\",\"math.NT\"]","methods":"[\"Generative Adversarial Network\"]","has_code":false}