Supersolvable Arrangement

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

↑ Stanley, Richard P. (1972). "Supersolvable lattices". Algebra Universalis 2: 197–217. doi:10.1007/BF02945028.
↑ 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.
↑ Yuzvinsky, Sergey (2001). "Orlik–Solomon algebras in algebra and topology". Russian Mathematical Surveys 56 (2): 293–364. doi:10.1070/RM2001v056n02ABEH000383. Bibcode: 2001RuMaS..56..293Y.

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[1] Stanley, Richard P. (1972). "Supersolvable lattices". Algebra Universalis 2: 197–217. doi:10.1007/BF02945028.

[2] 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.

[3] Yuzvinsky, Sergey (2001). "Orlik–Solomon algebras in algebra and topology". Russian Mathematical Surveys 56 (2): 293–364. doi:10.1070/RM2001v056n02ABEH000383. Bibcode: 2001RuMaS..56..293Y.