In mathematics, a supersolvable arrangement is a

hyperplane arrangement In geometry and combinatorics, an arrangement of hyperplanes is an arrangement of a finite set ''A'' of hyperplanes in a linear, affine, or projective space ''S''. Questions about a hyperplane arrangement ''A'' generally concern geometrical, top ...

which has a maximal

flag A flag is a piece of fabric (most often rectangular or quadrilateral) with a distinctive design and colours. It is used as a symbol, a signalling device, or for decoration. The term ''flag'' is also used to refer to the graphic design empl ...

with only

modular Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...

elements. Equivalently, the intersection semilattice of the arrangement is a supersolvable

lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an ornam ...

, in the sense of

Richard P. Stanley Richard Peter Stanley (born June 23, 1944) is an Emeritus Professor of Mathematics at the Massachusetts Institute of Technology, in Cambridge, Massachusetts. From 2000 to 2010, he was the Norman Levinson Professor of Applied Mathematics. He r ...

. 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 ...

, 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 refl ...

s of type A and B. It is known that the Orlik–Solomon algebra of a supersolvable arrangement is a

Koszul algebra In abstract algebra, a Koszul algebra R is a graded k-algebra over which the ground field k has a linear minimal graded free resolution, ''i.e.'', there exists an exact sequence: :\cdots \rightarrow R(-i)^ \rightarrow \cdots \rightarrow R(-2)^ ...

; 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 ''Uspekhi Matematicheskikh Nauk'' (russian: Успехи математических наук) is a Russian mathematical journal, published by the Russian Academy of Sciences and Moscow Mathematical Society and translated into English as ''Russia ...

, volume= 56 , year=2001, issue= 2, pages= 293–364, mr=1859708, doi=10.1070/RM2001v056n02ABEH000383, bibcode= 2001RuMaS..56..293Y

References

Discrete geometry Matroid theory