📐Discrete Geometry Unit 6 – Arrangements of Hyperplanes

Key Concepts and Definitions

• Hyperplanes are subspaces of dimension one less than the ambient space (e.g., planes in 3D space, lines in 2D space)
• An arrangement of hyperplanes is a finite set of hyperplanes in a vector space
  • Arrangements can be in any dimension, but most commonly studied in 2D and 3D
• Regions are the connected components of the complement of the union of hyperplanes
  • Bounded regions are called chambers, while unbounded regions are called unbounded chambers or cells
• Faces are the intersections of hyperplanes, including the empty set and the whole space
  • Faces form a partially ordered set (poset) ordered by reverse inclusion
• The intersection poset is the set of all faces ordered by reverse inclusion
• The characteristic polynomial is a polynomial encoding combinatorial information about the arrangement
• A central arrangement is one where all hyperplanes pass through a common point (the origin)

Geometric Foundations

• Hyperplanes in a vector space are characterized by linear equations of the form $a_1x_1 + a_2x_2 + ... + a_nx_n = b$
• The normal vector of a hyperplane is the vector $(a_1, a_2, ..., a_n)$ perpendicular to the hyperplane
• Two hyperplanes are parallel if their normal vectors are scalar multiples of each other
• The intersection of two non-parallel hyperplanes is a hyperplane of one lower dimension
• The angle between two hyperplanes can be calculated using the dot product of their normal vectors
• Projective geometry provides a natural setting for studying arrangements by adding points at infinity
• Oriented matroids capture the combinatorial essence of hyperplane arrangements without explicit coordinates

Types of Hyperplane Arrangements

• Central arrangements have all hyperplanes passing through a common point (usually the origin)
  • In a central arrangement, the intersection poset has a unique minimal element
• Affine arrangements are arrangements in affine space, where parallel hyperplanes are allowed
• Simplicial arrangements are arrangements where every chamber is a simplex
  • The braid arrangement is a well-known example of a simplicial arrangement
• Reflection arrangements arise from finite reflection groups (Coxeter groups)
  • Examples include the Coxeter arrangement of type A (braid arrangement) and type B
• Supersolvable arrangements have a special structure that allows for easier computation of invariants
• Complex arrangements are arrangements of complex hyperplanes in complex vector spaces
• Oriented matroids provide a combinatorial generalization of hyperplane arrangements

Combinatorial Properties

• The face poset encodes the combinatorial structure of the arrangement
  • The Möbius function of the face poset is related to the characteristic polynomial
• The intersection poset is the poset of all intersections of hyperplanes ordered by reverse inclusion
• The characteristic polynomial is a polynomial encoding combinatorial information
  • It can be computed using the Möbius function of the intersection poset
• The number of regions (chambers) is given by evaluating the characteristic polynomial at $-1$
• Zaslavsky's theorem relates the number of regions to the Möbius function of the intersection poset
• The Whitney numbers of the first and second kind count the number of faces of each dimension
• Hyperplane arrangements can be used to construct zonotopes, which are special polytopes
• The Orlik-Solomon algebra is a graded algebra encoding the combinatorics of the arrangement

Algebraic Aspects

• Each hyperplane arrangement gives rise to a module over a polynomial ring
  • The module structure encodes algebraic properties of the arrangement
• The Orlik-Terao algebra is a commutative algebra associated with a hyperplane arrangement
• The freeness of the module is related to the supersolvability of the arrangement
• Free arrangements have a basis of derivations, leading to a simple description of the module
• The Poincaré polynomial of the Orlik-Solomon algebra is related to the characteristic polynomial
• D-modules and perverse sheaves provide a deeper algebraic perspective on arrangements
• Arrangements can be studied using techniques from commutative algebra and algebraic geometry
• Resonance varieties are algebraic varieties encoding the first cohomology of the Orlik-Solomon algebra

Applications in Other Fields

• Hyperplane arrangements arise naturally in many areas of mathematics and physics
• In combinatorics, arrangements are used to study partition lattices, permutations, and graph colorings
• In topology, complements of arrangements are interesting spaces with nontrivial fundamental groups
  • The braid arrangement is closely related to the configuration space of distinct points in the plane
• In representation theory, reflection arrangements are related to root systems and Weyl groups
• In algebraic geometry, arrangements are used to construct wonderful compactifications and resolution of singularities
• In physics, arrangements appear in the study of Landau singularities and Feynman integrals
• Oriented matroids, which generalize arrangements, have applications in optimization and computational geometry
• Arrangements have been used to construct interesting examples in discrete geometry, such as counterexamples to the Hirsch conjecture

Computational Techniques

• Many computational problems involving arrangements can be solved using techniques from computational geometry
• The incremental algorithm can be used to efficiently compute the face poset of an arrangement
• The sweeping hyperplane method can be used to compute the intersection poset and characteristic polynomial
• Randomized algorithms, such as the Clarkson-Shor algorithm, can be used for efficient computation of arrangements
• Symbolic computation software, such as Macaulay2 and Sage, can be used to study arrangements
• Gröbner basis techniques are useful for studying the algebraic aspects of arrangements
• Discrete Morse theory can be used to study the topology of arrangement complements
• Efficient data structures, such as the doubly connected edge list, are used to represent arrangements in computer implementations

Advanced Topics and Open Problems

• The Terao conjecture states that the freeness of an arrangement depends only on its combinatorial structure (intersection poset)
  • The conjecture is known to hold for certain classes of arrangements but remains open in general
• The Orlik-Terao algebra is not yet fully understood, and its properties are the subject of ongoing research
• The relationship between resonance varieties and characteristic varieties is an active area of research
• Arrangements over finite fields have been studied in recent years and have connections to coding theory
• Infinite arrangements, where the number of hyperplanes is infinite, pose additional challenges and have been studied in special cases
• Arrangements in other contexts, such as hyperbolic or spherical geometry, have been investigated
• The topology of arrangement complements in higher dimensions is not fully understood and is an active area of research
• Arrangements with additional structure, such as multiarrangements or weighted arrangements, have been studied and pose new challenges