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Supersolvable hyperplane arrangements and matroids are known to give rise to certain Koszul algebras, namely their Orlik–Solomon algebras and graded Varchenko–Gel’fand algebras. We explore how this interacts with group actions, particularly for the braid arrangement and the action of the symmetric group, where the Hilbert functions of the algebras and their Koszul duals are given by Stirling numbers of the first and second kinds, respectively. The corresponding symmetric group representations exhibit branching rules that interpret Stirling number recurrences, which are shown to apply to all supersolvable arrangements. They also enjoy representation stability properties that follow from Koszul duality.
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This content will become publicly available on July 1, 2026
Geometric Aspects of the Jacobian of a Hyperplane Arrangement
Abstract An embedding of the complete bipartite graph $$K_{3,3}$$ in $$\mathbb{P}^{2}$$ gives rise to both a line arrangement and a bar-and-joint framework. For a generic placement of the six vertices, the graded Betti numbers of the logarithmic module of derivations of the line arrangement are constant, but an important example due to Ziegler shows that the graded Betti numbers are different when the points lie on a conic. Similarly, in rigidity theory a generic embedding of $$K_{3,3}$$ in the plane is an infinitesimally rigid bar-and-joint framework, but the framework is infinitesimally flexible when the points lie on a conic. We develop the theory of weak perspective representations of hyperplane arrangements to formalize and generalize the striking connection between hyperplane arrangements and rigidity theory that the example above suggests. In characteristic zero we show that there is a one-to-one correspondence between weak perspective representations of a hyperplane arrangement and polynomials of minimal degree in certain saturations of the Jacobian ideal of the arrangement, providing a connection to algebra. In this setting we can use duality theorems to explain how rigidity theory is reflected in the graded Betti numbers of the module of logarithmic derivations of a line arrangement.
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- Award ID(s):
- 2344588
- PAR ID:
- 10627279
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2025
- Issue:
- 13
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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