In mathematics, a supersolvable arrangement is a hyperplane arrangement that has a maximal flag consisting of modular elements. Equivalently, the intersection semilattice of the arrangement is a supersolvable lattice, in the sense of Richard P. Stanley.[1] As shown by Hiroaki Terao, a complex hyperplane arrangement is supersolvable if and only if its complement is fiber-type.[2]
Examples include arrangements associated with Coxeter groups of type A and B.
The Orlik–Solomon algebra of every supersolvable arrangement is a Koszul algebra; whether the converse is true is an open problem.[3]
References
edit- ^ Stanley, Richard P. (1972). "Supersolvable lattices". Algebra Universalis. 2: 197–217. doi:10.1007/BF02945028. MR 0309815. S2CID 189844197.
- ^ Terao, Hiroaki (1986). "Modular elements of lattices and topological fibration". Advances in Mathematics. 62 (2): 135–154. doi:10.1016/0001-8708(86)90097-6. MR 0865835.
- ^ Yuzvinsky, Sergey (2001). "Orlik–Solomon algebras in algebra and topology". Russian Mathematical Surveys. 56 (2): 293–364. Bibcode:2001RuMaS..56..293Y. doi:10.1070/RM2001v056n02ABEH000383. MR 1859708.