Algebraic combinatorics
Encyclopedia
Algebraic combinatorics is an area of mathematics
that employs methods of abstract algebra
, notably group theory
and representation theory
, in various combinatorial
contexts and, conversely, applies combinatorial techniques to problems in algebra
.
Algebraic Combinatorics 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. The book Algebraic Combinatorics by Bannai and Ito was published in 1983.
Through the early or mid 1990s, typical combinatorial objects of interest in algebraic combinatorics either admitted a lot of symmetries (association scheme
s, strongly regular graph
s, posets with a group action
) or possessed a rich algebraic structure, frequently of representation theoretic origin (symmetric function
s, Young tableaux). This period is reflected in the area 05E, Algebraic combinatorics, of the AMS
Mathematics Subject Classification
, introduced in 1991.
However, within the last decade or so, algebraic combinatorics came to be seen more expansively as an area of mathematics where the interaction of combinatorial and algebraic methods is particularly strong and significant. Thus the combinatorial topics may be enumerative
in nature or involve matroid
s, polytope
s, partially ordered set
s, or finite geometries
. On the algebraic side, besides group and representation theory, lattice theory and commutative algebra
are common. One of the fastest developing subfields within algebraic combinatorics is combinatorial commutative algebra
. Journal of Algebraic Combinatorics
, published by Springer-Verlag, is an international journal intended as a forum for papers in the field.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
that employs methods of abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, notably group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
and representation theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
, in various combinatorial
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
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 groups, rings, fields, modules, vector spaces, and algebras...
.
Algebraic Combinatorics 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. The book Algebraic Combinatorics by Bannai and Ito was published in 1983.
Through the early or mid 1990s, typical combinatorial objects of interest in algebraic combinatorics either admitted a lot of symmetries (association scheme
Association scheme
The theory of association schemes arose in statistics, in the theory of experimental design for the analysis of variance. In mathematics, association schemes belong to both algebra and combinatorics. Indeed, in algebraic combinatorics, association schemes provide a unified approach to many topics,...
s, strongly regular graph
Strongly regular graph
In graph theory, a discipline within mathematics, a strongly regular graph is defined as follows. Let G = be a regular graph with v vertices and degree k...
s, posets with a group action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. 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 transformations of the set...
) or possessed a rich algebraic structure, frequently of representation theoretic origin (symmetric function
Symmetric function
In algebra and in particular in algebraic combinatorics, the ring of symmetric functions, is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity...
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 mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...
Mathematics Subject Classification
Mathematics Subject Classification
The Mathematics Subject Classification is an alphanumerical classification scheme collaboratively produced by staff of and based on the coverage of the two major mathematical 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 an 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 that deals with 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 elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...
s, partially ordered set
Partially ordered set
In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...
s, or finite geometries
Finite geometry
A finite geometry is any geometric system that has only a finite number of points.Euclidean geometry, for example, is not finite, because a Euclidean line contains infinitely many points, in fact as many 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 ideals, and modules 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 mathematical 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.- External links :*...
, published by Springer-Verlag, is an international journal intended as a forum for papers in the field.