研究生課程《代數(shù)組合引論》教學(xué)大綱

研究生課程《代數(shù)組合引論》教學(xué)大綱課程編號：Math2081課程名稱：代數(shù)組合引論英文名稱：IntroductiontoAlgebraicCombinatorics

開課單位：數(shù)學(xué)科學(xué)學(xué)院開課學(xué)期：秋課內(nèi)學(xué)時：36教學(xué)方式：英文講授適用專業(yè)及層次：數(shù)學(xué)科學(xué)學(xué)院基礎(chǔ)數(shù)學(xué)專業(yè)碩士考核方式：考試預(yù)修課程：高等代數(shù)，圖論，近世代數(shù)[IntroductiontoAlgebraicCombinatorics]

代數(shù)組合引論

一．教學(xué)目標(biāo)與要求Thiscourseisaimedatgivingbasicknowledgeabout(i)representations

offinitegroupsand(ii)associationschemes.Studentsarerequiredto

havebasicknowledgeaboutlinearalgebraandaboutgroups,rings,fields.

二．課程內(nèi)容與學(xué)時分配Preface:introduction:anoverviewonalgebraiccombinatorics(2課時)Chapter1grouprepresentations(8

課時)1.1completereducibility1.2orthogonalityrelationsofcharacters1.3inducedrepresentations1.4applications:permutationrepresentations

Chapter2semi-simplealgebras(6課時)2.1Wedderburn'stheorem2.2grouprings:grouprepresentationsrevisited2.3Heckealgebrasoftransitivepermutationgroups

Chapter3associationschemes(8課時)3.1Bose-Mesneralgebra:the1st/2ndeigenmatrix3.2parameterrelations:theintersectionmatrices3.3Terwilligeralgebra3.4primitivity:sub-schemesandquotientschemes

Chapter4HammingschemesandJohnsonschemes(4課時)4.1theHammingschemeH(n,q)4.2theJohnsonschemeJ(v,k)

Chapter5P/Q-polynomialschemes(8課時)5.1Distance-regulargraphs(P-polynomialschemes)andtheirduals(Q-polynomialschemes):orthogonalpolynomials5.2Mooregraphs5.3LeonardpairsandAskey-Wilsonpolynomials5.4Towardstheclassificationof(PandQ)-polynomialschemes:thelistofknownexamples

四．教材E.BannaiandT.Ito,AlgebraicCombinatoricsI:AssociationSchemes,Benjamin/Cummings,MenloPark,California,1984.

四、

主要參考書1．A.E.Brouwer,A.M.Cohen,A.Neumaier,Dista