arXiv:2611.13050v1 Announce Type: new Abstract: For real affine hyperplane arrangements, we prove Lam's determinant conjecture for the inverse Varchenko matrix restricted to bounded chambers. The proof combines Lam's comparison of twisted intersection forms with the period determinant formula of Varchenko, Loeser-Sabbah and Douai-Terao. For normal crossing arrangements, we introduce the taut path algebra and the chamber intersection algebra and establish a Koszul duality between them, yielding an explicit multivariate formula for the inverse Varchenko matrix. After specializing the parameters to $q$, its restriction to bounded chambers is a scalar multiple of the $q$-intersection matrix, proving a determinant conjecture of the authors. We also discuss how the Koszul duality results generalize to affine oriented matroids.
Continue to arXiv Mathematics →You are being forwarded in 6s — stay on this page