Algebraic combinatorics

The Fano matroid, derived from the Fano plane. Matroids are one of many areas studied in 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.

History

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.

Scope

Algebraic combinatorics has come 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 matroids, polytopes, partially ordered sets, or finite geometries. On the algebraic side, besides group and representation theory, lattice theory and commutative algebra are common. The Journal of Algebraic Combinatorics, published by Springer-Verlag, is an international journal intended as a forum for papers in the field.

Important topics

Symmetric functions

{{|Ring of symmetric functions}} The ring of symmetric functions is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity. This ring serves as universal structure in which relations between symmetric polynomials can be expressed in a way independent of the number n of indeterminates (but its elements are neither polynomials nor functions). Among other things, this ring plays an important role in the representation theory of the symmetric groups.

See also

• Algebraic graph theory
• Combinatorial commutative algebra
• Polyhedral combinatorics

References

• Bannai, Eiichi; Ito, Tatsuro (1984). Algebraic combinatorics I: Association schemes. Menlo Park, CA: The Benjamin/Cummings Publishing Co., Inc. pp. xxiv+425. ISBN 0-8053-0490-8. MR 0882540. {{cite book}}: Cite has empty unknown parameter: |1= (help)
• Godsil, C. D. (1993). Algebraic Combinatorics. New York: Chapman and Hall. ISBN 0-412-04131-6. MR 1220704.
• 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
• Doron Zeilberger, Enumerative and Algebraic Combinatorics