Seminaire Lotharingien de Combinatoire Article B58c

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Niveau: Supérieur, Doctorat, Bac+8
Seminaire Lotharingien de Combinatoire 58 (2008), Article B58c OPERADS AND ALGEBRAIC COMBINATORICS OF TREES F. CHAPOTON Abstract. This text is an introduction to operads from a combinatorial point of view. We give the definition, several examples of operads and the construction of a group from each operad. 1. Introduction Combinatorics is concerned with finite objects, and considers problems such as enumerating some finite sets or finding bijections between different sets. Algebraic combinatorics could be defined as the reciprocal application of methods and ideas of algebra to combinatorics or the other way round. Since the pioneering work of Rota “on the foundations of combinatorial theory”, these subjects have become an active and central piece of mathematics. Up to now, most of the algebraic structures used in algebraic combinatorics have been well-established and classical notions of algebra. One of the main streams is the use of associative algebras and their representation theory, where combina- torial objects appear typically as labelling sets for some basis, or in a description of some modules. For instance, partitions, as considered in number theory, play an important role in representation theory of symmetric groups, general linear groups and many related algebras. Similarly, there has been recently a lot of activity on relations between Hopf algebras and combinatorial objects. One salient point of these works is the ap- pearance of new kinds of combinatorial entities, such as different kinds of trees, in an algebraic context.

  • algebras

  • symmetric operad

  • r's then composing

  • top compositions

  • algebras up

  • associative algebras

  • permutations ?? associative

  • composition maps

  • trees


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Se´minaireLotharingiendeCombinatoire58(2008),ArticleB58cOPERADSANDALGEBRAICCOMBINATORICSOFTREESF.CHAPOTONAbstract.Thistextisanintroductiontooperadsfromacombinatorialpointofview.Wegivethedefinition,severalexamplesofoperadsandtheconstructionofagroupfromeachoperad.1.IntroductionCombinatoricsisconcernedwithfiniteobjects,andconsidersproblemssuchasenumeratingsomefinitesetsorfindingbijectionsbetweendifferentsets.Algebraiccombinatoricscouldbedefinedasthereciprocalapplicationofmethodsandideasofalgebratocombinatoricsortheotherwayround.SincethepioneeringworkofRota“onthefoundationsofcombinatorialtheory”,thesesubjectshavebecomeanactiveandcentralpieceofmathematics.Uptonow,mostofthealgebraicstructuresusedinalgebraiccombinatoricshavebeenwell-establishedandclassicalnotionsofalgebra.Oneofthemainstreamsistheuseofassociativealgebrasandtheirrepresentationtheory,wherecombina-torialobjectsappeartypicallyaslabellingsetsforsomebasis,orinadescriptionofsomemodules.Forinstance,partitions,asconsideredinnumbertheory,playanimportantroleinrepresentationtheoryofsymmetricgroups,generallineargroupsandmanyrelatedalgebras.Similarly,therehasbeenrecentlyalotofactivityonrelationsbetweenHopfalgebrasandcombinatorialobjects.Onesalientpointoftheseworksistheap-pearanceofnewkindsofcombinatorialentities,suchasdifferentkindsoftrees,inanalgebraiccontext.Thepresenttextwouldliketoadvocateforthepotentialuseincombinatoricsofamorerecentalgebraicnotion,whichiscalledanoperad.Thisnotionhasfirstbeenintroducedinalgebraictopologyintheearly1970’s(Boardman–Vogt,May[BV73,May72])forthestudyofloopspaces.Therewasa“renaissance”,arenewedinterestinoperads,inthe1990’s,withtheKoszuldualitytheoryintroducedbyGinzburgandKapranov[GK94]andalsoinrelationwithmodulispacesofcurves[LSV97].Morerecently,thetheoryofoperadshasseenfurtherdevelopmentsinmanydirections.Operadsareusefultodescribeandworkwithcomplicatednewkindsofalgebrasandalgebrasuptohomotopy.
2F.CHAPOTONTheideaofanoperadiscloselyrelatedtotrees,inasimilarsensethattheideaofanassociativealgebraisrelatedtothenotionofwords.Operadscouldbeusefulincombinatoricsinseveralways.First,theycanshedlightonsomeclassicalcombinatorialobjects,byendowingthemwithnewalgebraicstructures,muchlikeassociativealgebrasdoforpartitions.Next,theycanprovidethesettingforveryrefinedversionsofgeneratingseries,wheretheindicesarenolongerthesetofintegers,butsomesetofcombinatorialobjects,suchastreesofvarioussorts.Wehavetriedtopresentthetheoryofoperadstoacombinatorialaudience,byaspecificchoiceofillustrativeexamples.ThecombinatorialistwhoalreadyknowssomethingaboutthecombinatorialtheoryofspeciesofstructuresofJoyal[Joy86](whichaimsatprovidingawaytodealwithgeneratingseriesinsomeuniversalway)willseeitappearfromanotherpointofview.Letusgivesomefurthergeneralreferencesonoperads:ageneralpresentation[Lod96]andtwomonographs[MSS02,Smi01].Thistextisthewrittenversionoflecturesgivenatthe58thSe´minaireLotha-ringiendeCombinatoireinMarch2007.Wegiveseveralequivalentdefinitionsofoperads,examplesofoperadsbasedoncombinatorialobjects,andexplainhowonecanbuildagroupof“invertibleformalpowerseries”startingfromanoperad.2.MotivationThissectionexplainsbrieflysomepossiblemotivationsforoperadsincombina-torics.2.1.Wordsandpermutations.Letusrecalltheclassicalrelationbetweenwordsandassociativity.AnalphabetAisasetofletters{a,b,c,...}.AwordwinthealphabetAisasequenceoflettersw=(w1,w2,...,wk).Thereisabasicoperationonwordsgivenbyconcatenation,whichisassociative.Infact,thesetofwordsisexactlythefreeassociativemonoidonthesetA.Sothestudyofwordsnaturallytakesplaceinthesettingofassociativealgebras.Considernowthealphabet{a1,...,an}.Thenthesetofwordswhereeachletteraiappearsexactlyoncecanbeseenasthesetofpermutationsof{1,...,n}.Inanaturalsense,permutationsencodeallpossibleoperationsthatcanbemadewithndistinctelementsinanassociativealgebra.Onecanfindasimilarrelationbetweensomekindsoftreesandsomenewkindsofalgebraicstructures.Thereisaparallelbetweenwordsorpermutations←→associativealgebras,rootedtrees←→pre-Liealgebras,planarbinarytrees←→dendriformalgebras.