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Algebraic combinatorics

 

 
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Algebraic combinatorics
 
 
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Algebraic combinatorics is an area of mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
 that employs methods of abstract algebra
Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as group , ring , field , module , vector spaces, and algebra over a field....
, notably group theory
Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as group .The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring , field , and vector spaces can all be seen as groups endowed with additional operations and axioms....
 and representation theory
Representation theory

Representation theory is a branch of mathematics that studies abstract algebra algebraic structures by representing their element as linear transformations of vector spaces....
, in various combinatorial
Combinatorics

Combinatorics is a branch of pure mathematics concerning the study of Countable set objects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics....
 contexts and, conversely, applies combinatorial techniques to problems in algebra
Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as group , ring , field , module , vector spaces, and algebra over a field....
. It is one of the youngest combinatorial disciplines. Thus, in the preface to his Combinatorial Theory, published in 1979, Martin Aigner wrote about "growing consensus that combinatorics should be divided into three parts" (Enumeration, Order theory, Configurations), without even mentioning algebraic combinatorics by name.

Through the early or mid 1990s, typical combinatorial objects of interest in algebraic combinatorics either admitted a lot of symmetries (association scheme
Association scheme

In mathematics, association schemes are structures that appear in many different forms in the fields of combinatorics and statistics....
s, strongly regular graph
Strongly regular graph

Let G = be a regular graph with v vertices and degree k. G is said to be strongly regular if there are also integers ? and ? such that:...
s, posets with a group action
Group action

In algebra and geometry, a group action is a way of describing symmetry of objects using group . The essential elements of the object are described by a Set and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformation of the set....
) or possessed a rich algebraic structure, frequently of representation theoretic origin (symmetric function
Symmetric function

In mathematics, the term "symmetric function" can mean two different things. A symmetric function of n variables is one whose value at any n-tuple of arguments is the same as its value at any permutation of that n-tuple....
s, Young tableaux).






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Encyclopedia


Algebraic combinatorics is an area of mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
 that employs methods of abstract algebra
Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as group , ring , field , module , vector spaces, and algebra over a field....
, notably group theory
Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as group .The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring , field , and vector spaces can all be seen as groups endowed with additional operations and axioms....
 and representation theory
Representation theory

Representation theory is a branch of mathematics that studies abstract algebra algebraic structures by representing their element as linear transformations of vector spaces....
, in various combinatorial
Combinatorics

Combinatorics is a branch of pure mathematics concerning the study of Countable set objects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics....
 contexts and, conversely, applies combinatorial techniques to problems in algebra
Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as group , ring , field , module , vector spaces, and algebra over a field....
. It is one of the youngest combinatorial disciplines. Thus, in the preface to his Combinatorial Theory, published in 1979, Martin Aigner wrote about "growing consensus that combinatorics should be divided into three parts" (Enumeration, Order theory, Configurations), without even mentioning algebraic combinatorics by name.

Through the early or mid 1990s, typical combinatorial objects of interest in algebraic combinatorics either admitted a lot of symmetries (association scheme
Association scheme

In mathematics, association schemes are structures that appear in many different forms in the fields of combinatorics and statistics....
s, strongly regular graph
Strongly regular graph

Let G = be a regular graph with v vertices and degree k. G is said to be strongly regular if there are also integers ? and ? such that:...
s, posets with a group action
Group action

In algebra and geometry, a group action is a way of describing symmetry of objects using group . The essential elements of the object are described by a Set and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformation of the set....
) or possessed a rich algebraic structure, frequently of representation theoretic origin (symmetric function
Symmetric function

In mathematics, the term "symmetric function" can mean two different things. A symmetric function of n variables is one whose value at any n-tuple of arguments is the same as its value at any permutation of that n-tuple....
s, Young tableaux). This period is reflected in the area 05E, Algebraic combinatorics, of the AMS
American Mathematical Society

The American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematics research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians....
 Mathematics Subject Classification
Mathematics Subject Classification

The Mathematics Subject Classification is an alphanumerical classification scheme formulated by the American Mathematical Society based on the coverage of two major reviewing databases Mathematical Reviews and Zentralblatt MATH....
, introduced in 1991. However, within the last decade or so, algebraic combinatorics came to be seen more expansively as the area of mathematics where the interaction of combinatorial and algebraic methods is particularly strong and significant. Thus the combinatorial topics may be enumerative
Enumerative combinatorics

Enumerative combinatorics is an area of Combinatorics on the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations....
 in nature or involve matroid
Matroid

In combinatorics, a branch of mathematics, a matroid or independence structure is a structure that captures the essence of a notion of "independence" that generalizes linear independence in vector spaces....
s, polytope
Polytope

In geometry, polytope is a generic term that can refer to a two-dimensional polygon, a three-dimensional polyhedron, or any of the various generalizations thereof, including generalizations to higher dimensions and other abstractions ....
s, partially ordered set
Partially ordered set

In mathematics, especially order theory, a partially ordered set formalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set ....
s, or finite geometries
Finite geometry

A finite geometry is any geometry system that has only a finite set number of point .Euclidean geometry, for example, is not finite, because a Euclidean line contains infinitely many points, in fact precisely the same number of points as there are real numbers....
. On the algebraic side, besides group and representation theory, lattice theory and commutative algebra
Commutative algebra

Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideal , and module over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra....
 are common. One of the fastest developing subfields within algebraic combinatorics is combinatorial commutative algebra
Combinatorial commutative algebra

Combinatorial commutative algebra is a relatively new, rapidly developing mathematics discipline. As the name implies, it lies at the intersection of two more established fields, commutative algebra and combinatorics, and frequently uses methods of one to address problems arising in the other....
. Journal of Algebraic Combinatorics
Journal of Algebraic Combinatorics

Journal of Algebraic Combinatorics is an international mathematical journal dedicated to the field of algebraic combinatorics. It is published bimonthly by Springer-Verlag....
, published by Springer-Verlag, is an international journal intended as a forum for papers in the field.

See also


  • Algebraic graph theory
    Algebraic graph theory

    Algebraic graph theory is a branch of mathematics in which algebra methods are applied to problems about Graph .In one sense, algebraic graph theory studies graphs in connection with linear algebra....
  • Spectral graph theory
    Spectral graph theory

    In mathematics, spectral graph theory is the study of properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of its adjacency matrix or Laplacian matrix....