Orlik–Solomon sheaf homology on a geometric lattice concentrates in top degree and decomposes as a sum of local OS algebras tensored with top homology of complementary geometric semilattices.
Hamiltonian Cycles in Simplicial and Supersolvable Hyperplane Arrangements
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abstract
Motivated by the Gray code interpretation of Hamiltonian cycles in Cayley graphs, we investigate the existence of Hamiltonian cycles in tope graphs of hyperplane arrangements, with a focus on simplicial, reflection, and supersolvable arrangements. We confirm Hamiltonicity for all 3-dimensional simplicial arrangements listed in the Gr\"unbaum--Cuntz catalogue. Extending earlier results by Conway, Sloane, and Wilks, we prove that all restrictions of finite reflection arrangements, including all Weyl groupoids and crystallographic arrangements, admit Hamiltonian cycles. Finally, we further establish that all supersolvable hyperplane arrangements and supersolvable oriented matroids have Hamiltonian cycles, offering a constructive proof based on their inductive structure.
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All restrictions of reflection arrangements are gamma-positive; type B Chow polynomials admit an explicit combinatorial formula, and intermediate type D restrictions interpolate arithmetically between B and D invariants.
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Orlik--Solomon sheaf homology of geometric lattices
Orlik–Solomon sheaf homology on a geometric lattice concentrates in top degree and decomposes as a sum of local OS algebras tensored with top homology of complementary geometric semilattices.
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On gamma-vectors and Chow polynomials of restrictions of reflection arrangements
All restrictions of reflection arrangements are gamma-positive; type B Chow polynomials admit an explicit combinatorial formula, and intermediate type D restrictions interpolate arithmetically between B and D invariants.