Algebraic combinatorics

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. 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 schemes, strongly regular graphs, posets with a group action) or possessed a rich algebraic structure, frequently of representation theoretic origin (symmetric functions, 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 the 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 matroids, polytopes, partially ordered sets, 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.

References

* Takayuki Hibi, Algebraic combinatorics on convex polytopes, Carslaw Publications, Glebe, Australia, 1992

* Melvin Hochster, Cohen-Macaulay rings, combinatorics, and simplicial complexes. Ring theory, II (Proc. Second Conf., Univ. Oklahoma, Norman, Okla., 1975), pp. 171--223. Lecture Notes in Pure and Appl. Math., vol. 26, Dekker, New York, 1977.

* Ezra Miller, Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, NY, 2005. ISBN 0-387-22356-8

* Richard Stanley, Combinatorics and commutative algebra. Second edition, Progress in Mathematics, vol. 41. Birkhäuser, Boston, MA, 1996. ISBN 0-8176-3836-9

* Bernd Sturmfels, Gröbner bases and convex polytopes, University Lecture Series, vol. 8, American Mathematical Society, Providence, RI, 1996. ISBN 0-8218-0487-1

Combinatorics