In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
and
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
, an arrangement of hyperplanes is an
arrangement
In music, an arrangement is a musical adaptation of an existing composition. Differences from the original composition may include reharmonization, melodic paraphrasing, orchestration, or formal development. Arranging differs from orches ...
of a finite set ''A'' of
hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
s in a
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
,
affine
Affine may describe any of various topics concerned with connections or affinities.
It may refer to:
* Affine, a relative by marriage in law and anthropology
* Affine cipher, a special case of the more general substitution cipher
* Affine comb ...
, or
projective space ''S''.
Questions about a hyperplane arrangement ''A'' generally concern geometrical, topological, or other properties of the complement, ''M''(''A''), which is the set that remains when the hyperplanes are removed from the whole space. One may ask how these properties are related to the arrangement and its intersection semilattice.
The intersection
semilattice
In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has a ...
of ''A'', written ''L''(''A''), is the set of all
subspaces that are obtained by intersecting some of the hyperplanes; among these subspaces are ''S'' itself, all the individual hyperplanes, all intersections of pairs of hyperplanes, etc. (excluding, in the affine case, the empty set). These intersection subspaces of ''A'' are also called the flats of ''A''. The intersection semilattice ''L''(''A'') is partially ordered by ''reverse inclusion''.
If the whole space ''S'' is 2-dimensional, the hyperplanes are
lines; such an arrangement is often called an
arrangement of lines
In music, an arrangement is a musical adaptation of an existing composition. Differences from the original composition may include reharmonization, melodic paraphrasing, orchestration, or formal development. Arranging differs from orchestr ...
. Historically, real arrangements of lines were the first arrangements investigated. If ''S'' is 3-dimensional one has an arrangement of planes.
General theory
The intersection semilattice and the matroid
The intersection semilattice ''L''(''A'') is a meet semilattice and more specifically is a
geometric semilattice. If the arrangement is linear or projective, or if the intersection of all hyperplanes is nonempty, the intersection lattice is a
geometric lattice
In the mathematics of matroids and lattices, a geometric lattice is a finite atomistic semimodular lattice, and a matroid lattice is an atomistic semimodular lattice without the assumption of finiteness. Geometric lattices and matroid lattices, r ...
.
(This is why the semilattice must be ordered by reverse inclusion—rather than by inclusion, which might seem more natural but would not yield a geometric (semi)lattice.)
When ''L''(''A'') is a lattice, the
matroid
In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being ...
of ''A'', written ''M''(''A''), has ''A'' for its ground set and has rank function ''r''(''S'') := codim(''I''), where ''S'' is any subset of ''A'' and ''I'' is the intersection of the hyperplanes in ''S''. In general, when ''L''(''A'') is a semilattice, there is an analogous matroid-like structure called a
semimatroid, which is a generalization of a matroid (and has the same relationship to the intersection semilattice as does the matroid to the lattice in the lattice case), but is not a matroid if ''L''(''A'') is not a lattice.
Polynomials
For a subset ''B'' of ''A'', let us define ''f''(''B'') := the intersection of the hyperplanes in ''B''; this is ''S'' if ''B'' is empty.
The characteristic polynomial of ''A'', written ''p
A''(''y''), can be defined by
:
summed over all subsets ''B'' of ''A'' except, in the affine case, subsets whose intersection is empty. (The dimension of the empty set is defined to be −1.) This polynomial helps to solve some basic questions; see below.
Another polynomial associated with ''A'' is the Whitney-number polynomial ''w
A''(''x'', ''y''), defined by
:
summed over ''B'' ⊆ ''C'' ⊆ ''A'' such that ''f''(''B'') is nonempty.
Being a geometric lattice or semilattice, ''L''(''A'') has a characteristic polynomial, ''p''
''L''(''A'')(''y''), which has an extensive theory (see
matroid
In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being ...
). Thus it is good to know that ''p''
''A''(''y'') = ''y''
''i'' ''p''
''L''(''A'')(''y''), where ''i'' is the smallest dimension of any flat, except that in the projective case it equals ''y''
''i'' + 1''p''
''L''(''A'')(''y'').
The Whitney-number polynomial of ''A'' is similarly related to that of ''L''(''A'').
(The empty set is excluded from the semilattice in the affine case specifically so that these relationships will be valid.)
The Orlik–Solomon algebra
The intersection semilattice determines another combinatorial invariant of the arrangement, the
Orlik–Solomon algebra. To define it, fix a commutative subring ''K'' of the base field and form the
exterior algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
''E'' of the vector space
:
generated by the hyperplanes.
A
chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or module (mathematics), modules) and a sequence of group homomorphism, homomorphisms between consecutive groups such that the image (mathemati ...
structure is defined on ''E'' with the usual boundary operator
.
The Orlik–Solomon algebra is then the quotient of ''E'' by the
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
generated by elements of the form
for which
have empty intersection, and by boundaries of elements of the same form for which
has
codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties.
For affine and projective algebraic varieties, the codimension equals the ...
less than ''p''.
Real arrangements
In
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
affine space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
, the complement is disconnected: it is made up of separate pieces called cells or regions or chambers, each of which is either a bounded region that is a
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope ...
polytope
In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -d ...
, or an unbounded region that is a convex
polyhedral region which goes off to infinity.
Each flat of ''A'' is also divided into pieces by the hyperplanes that do not contain the flat; these pieces are called the faces of ''A''.
The regions are faces because the whole space is a flat.
The faces of codimension 1 may be called the facets of ''A''.
The face semilattice of an arrangement is the set of all faces, ordered by ''inclusion''. Adding an extra top element to the face semilattice gives the face lattice.
In two dimensions (i.e., in the real affine
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
) each region is a convex
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
(if it is bounded) or a convex polygonal region which goes off to infinity.
* As an example, if the arrangement consists of three parallel lines, the intersection semilattice consists of the plane and the three lines, but not the empty set. There are four regions, none of them bounded.
* If we add a line crossing the three parallels, then the intersection semilattice consists of the plane, the four lines, and the three points of intersection. There are eight regions, still none of them bounded.
* If we add one more line, parallel to the last, then there are 12 regions, of which two are bounded
parallelogram
In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equa ...
s.
Typical problems about an arrangement in ''n''-dimensional real space are to say how many regions there are, or how many faces of dimension 4, or how many bounded regions. These questions can be answered just from the intersection semilattice. For instance, two basic theorems, from Zaslavsky (1975), are that the number of regions of an affine arrangement equals (−1)
''n''''p''
''A''(−1) and the number of bounded regions equals (−1)
''n''p
''A''(1). Similarly, the number of ''k''-dimensional faces or bounded faces can be read off as the coefficient of ''x''
''n''−''k'' in (−1)
''n'' w
''A'' (−''x'', −1) or (−1)
''n''''w''
''A''(−''x'', 1).
designed a fast algorithm to determine the face of an arrangement of hyperplanes containing an input point.
Another question about an arrangement in real space is to decide how many regions are
simplices
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
(the ''n''-dimensional generalization of
triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, an ...
s and
tetrahedra
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
). This cannot be answered based solely on the intersection semilattice. The
McMullen problem asks for the smallest arrangement of a given dimension in general position in
real projective space
In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space.
Basic properties Construction
A ...
for which there does not exist a cell touched by all hyperplanes.
A real linear arrangement has, besides its face semilattice, a
poset
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
of regions, a different one for each region. This poset is formed by choosing an arbitrary base region, ''B''
0, and associating with each region ''R'' the set ''S''(''R'') consisting of the hyperplanes that separate ''R'' from ''B''. The regions are partially ordered so that ''R''
1 ≥ ''R''
2 if ''S''(''R''
1, ''R'') contains ''S''(''R''
2, ''R''). In the special case when the hyperplanes arise from a
root system
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representati ...
, the resulting poset is the corresponding
Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections th ...
with the weak order. In general, the poset of regions is
ranked
A ranking is a relationship between a set of items such that, for any two items, the first is either "ranked higher than", "ranked lower than" or "ranked equal to" the second.
In mathematics, this is known as a weak order or total preorder of ...
by the number of separating hyperplanes and its
Möbius function
The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most oft ...
has been computed .
Vadim Schechtman and
Alexander Varchenko introduced a matrix indexed by the regions. The matrix element for the region
and
is given by the product of indeterminate variables
for every hyperplane H that separates these two regions. If these variables are specialized to be all value q, then this is called the q-matrix (over the Euclidean domain