Supersolvable arrangement explained

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

  1. 0309815. Stanley. Richard P.. Richard P. Stanley. Supersolvable lattices. Algebra Universalis. 2 . 1972. 197–217. 10.1007/BF02945028. 189844197.
  2. 0865835. Terao. Hiroaki. Hiroaki Terao. Modular elements of lattices and topological fibration. Advances in Mathematics. 62 . 1986. 2. 135–154. 10.1016/0001-8708(86)90097-6 . free.
  3. Sergey. Yuzvinsky. Orlik–Solomon algebras in algebra and topology. Russian Mathematical Surveys. 56 . 2001. 2. 293–364. 1859708. 10.1070/RM2001v056n02ABEH000383. 2001RuMaS..56..293Y.