In mathematics, a supersolvable arrangement is a
hyperplane arrangement that has a maximal
flag
A flag is a piece of textile, fabric (most often rectangular) with distinctive colours and design. It is used as a symbol, a signalling device, or for decoration. The term ''flag'' is also used to refer to the graphic design employed, and fla ...
consisting of
modular elements. Equivalently, the
intersection semilattice of the arrangement is a
supersolvable lattice
In mathematics, a supersolvable lattice is a graded poset, graded Lattice (order), lattice that has a maximal total order#Chains, chain of elements, each of which obeys a certain modularity relationship. The definition encapsulates many of the nice ...
, in the sense of
Richard P. Stanley. As shown by
Hiroaki Terao
is a Japanese mathematician, known as, with Peter Orlik and Louis Solomon, a pioneer of the theory of arrangements of hyperplanes. He was awarded a Mathematical Society of Japan Algebra Prize in 2010.
Education
Terao started his studies at the U ...
,
a complex hyperplane arrangement is supersolvable if and only if its complement is fiber-type.
Examples include arrangements associated with
Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...
s 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.
[{{cite journal, first=Sergey, last= Yuzvinsky, title= Orlik–Solomon algebras in algebra and topology, journal= Russian Mathematical Surveys, volume= 56 , year=2001, issue= 2, pages= 293–364, mr=1859708, doi=10.1070/RM2001v056n02ABEH000383, bibcode= 2001RuMaS..56..293Y]
References
Discrete geometry
Matroid theory