Algebraic combinatorics: Difference between revisions

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{{Short description|Area of combinatorics}}

{{for|the academic journal|Algebraic Combinatorics (journal)}}

[[File:fano plane.svg|thumb|The Fano [[matroid]], derived from the [[Fano plane]]. Matroids are one of many areaskinds of objects studied in '''algebraic combinatorics'''.]]

{{use dmy dates|date=January 2022}}

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==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 combinatorics|enumerative]] in nature or involve [[matroid]]s, [[polytope]]s, [[partially ordered set]]s, or [[finite geometry|finite geometries]]. On the algebraic side, besides group theory and representation theory, [[lattice theory]] and [[commutative algebra]] are commoncommonly used.

==Important topics==

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===Young tableaux===

{{main|Young tableau}}

A [[Young tableau]] (pl.: ''tableaux'') is a [[combinatorics|combinatorial]] object useful in [[representation theory]] and [[Schubert calculus]]. It provides a convenient way to describe the [[group representation]]s of the [[symmetric group|symmetric]] and [[general linear group|general linear]] groups and to study their properties. Young tableaux were introduced by [[Alfred Young (mathematician)|Alfred Young]], a [[mathematician]] at [[University of Cambridge|Cambridge University]], in 1900. They were then applied to the study of the symmetric group by [[Georg Frobenius]] in 1903. Their theory was further developed by many mathematicians, including [[Percy MacMahon]], [[W. V. D. Hodge]], [[Gilbert de Beauregard Robinson|G. de B. Robinson]], [[Gian-Carlo Rota]], [[Alain Lascoux]], [[Marcel-Paul Schützenberger]] and [[Richard P. Stanley]].

===Matroids===

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* [[Algebraic graph theory]]

* [[Combinatorial commutative algebra]]

* [[Polyhedral combinatorics]]

* [[Algebraic Combinatorics (journal)|''Algebraic Combinatorics'' (journal)]]

* ''[[Journal of Algebraic Combinatorics]]''

* [[International Conference on Formal Power Series and Algebraic Combinatorics]]

* [[Polyhedral combinatorics]]

== Citations ==

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| url = http://library.msri.org/books/Book38/index.html

| isbn = 052177087-4

| ref = none

}}

*{{Cite book| title = Algebraic combinatorics on convex polytopes

| last = Hibi | first = Takayuki | year = 1992

| publisher = Carslaw Publications | ___location = Glebe, Australia |isbn=1875399046 |oclc=29023080

| ref = none

}}

*{{Cite journal | title = Cohen-Macaulay rings, combinatorics, and simplicial complexes

| last = Hochster | first = Melvin

| author-link = Melvin Hochster

| journal = Lecture Notes in Pure and Applied Mathematics

| publisher = Dekker | ___location = New York

| year = 1975 | volume = 26 | pages = 171–223

| ref = none

}}

*{{cite conference |author-link=Melvin Hochster |first=Melvin |last=Hochster |title=Cohen–Macaulay rings, combinatorics, and simplicial complexes |book-title=Ring Theory II: Proceedings of the Second Oklahoma Conference |publisher=Dekker |series=Lecture Notes in Pure and Applied Mathematics |volume=26 |date=1977 |isbn=0-8247-6575-3 |pages=171–223 |oclc=610144046 |url=https://archive.org/details/ringtheoryiiproc0026ring |zbl=0351.13009}}

*{{Cite book| title = Combinatorial commutative algebra

| last1 = Miller | first1 = Ezra

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| author2-link = Bernd Sturmfels

| year = 2005

| publisher = Springer-Verlag | ___location = New York

| volume = 227 | series = [[Graduate Texts in Mathematics]]

| isbn = 0-387-22356-8 |zbl=1066.13001|url={{GBurl|OYBCAAAAQBAJ|pg=PR11}}

| ref = none

}}

*{{Cite book| title = Combinatorics and commutative algebra | edition = Second2nd

| last = Stanley | first = Richard P. | year = 1996

| author-link = Richard P. Stanley

| publisher = Birkhäuser | ___location = Boston

| volume = 41 | series = Progress in Mathematics

| isbn = 0-8176-3836-9 |zbl=0838.13008 |url={{GBurl|tZVCAAAAQBAJ|pg=PR5}}

| ref = none

}}

*{{cite book| title = Gröbner bases and convex polytopes

| last = Sturmfels | first = Bernd | year = 1996

| author-link = Bernd Sturmfels

| publisher = [[American Mathematical Society]] | ___location = Providence, RI

| volume = 8 | series = University Lecture Series

| url = https://archive.org/details/grobnerbasesconv0000stur | url-access = registration | via = [[Internet Archive]]

| isbn = 0-8218-0487-1 |oclc=907364245 |zbl=0856.13020

| ref = none

}}

*{{Cite book| chapter = Enumerative and Algebraic Combinatorics

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| publisher = Princeton University Press

| chapter-url = http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/enuPCM.pdf

| ref = none

}}

{{refend}}