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Title:
Character Theory and the Symplectic Group
Creator:
Ruiz, Joseph
Fleming, Courtney
Vigil, Gabriel
Place of Publication:
Denver, CO
Publisher:
Metropolitan State University of Denver
Publication Date:

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Genre:
Conference Papers ( sobekcm )

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Collected for Auraria Institutional Repository by the Self-Submittal tool. Submitted by Matthew Mariner.
General Note:
Faculty mentor: Amanda Schaeffer Fry
General Note:
Mentor: Mathematics

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Auraria Institutional Repository
Holding Location:
Auraria Library
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All applicable rights reserved by the source institution and holding location.

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CharacterTheoryandtheSymplecticGroup CharacterTheoryandtheSymplecticGroup CourtneyFleming 1 JosephRuiz 1 GabrielVigil 1 AlexanderTusa 1 Dr.MandiSchaeerFry 2 1 DepartmentofMathematicalandComputerSciences MetropolitanStateUniversityofDenver 2 FacultyAdvisor UndergraduateResearchConference,April2020

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CharacterTheoryandtheSymplecticGroup GroupDenition Denition Group :asetwithanoperation suchthat: closureforall a ; b 2 G wehave a b 2 G identity e a = a e = a associative a b c = a b c inverse a a )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 = a )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 a = e

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CharacterTheoryandtheSymplecticGroup GroupDenition Denition Group :asetwithanoperation suchthat: closureforall a ; b 2 G wehave a b 2 G identity e a = a e = a associative a b c = a b c inverse a a )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 = a )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 a = e

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CharacterTheoryandtheSymplecticGroup GroupDenition Denition Group :asetwithanoperation suchthat: closureforall a ; b 2 G wehave a b 2 G identity e a = a e = a associative a b c = a b c inverse a a )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 = a )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 a = e

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CharacterTheoryandtheSymplecticGroup GroupDenition Denition Group :asetwithanoperation suchthat: closureforall a ; b 2 G wehave a b 2 G identity e a = a e = a associative a b c = a b c inverse a a )]TJ/F27 7.9701 Tf 6.586 0 Td [(1 = a )]TJ/F27 7.9701 Tf 6.587 0 Td [(1 a = e

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CharacterTheoryandtheSymplecticGroup TheGroup S 3 S 3 isagroupthatrepresentsthesymmetriesofanequilateraltriangle

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CharacterTheoryandtheSymplecticGroup Groupactiononequilateraltriangle x y z x y z e

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CharacterTheoryandtheSymplecticGroup Groupactiononequilateraltriangle x y z y z x r 120

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CharacterTheoryandtheSymplecticGroup Groupactiononequilateraltriangle x y z z x y r 240

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CharacterTheoryandtheSymplecticGroup Groupactiononequilateraltriangle x y z x z y a

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CharacterTheoryandtheSymplecticGroup Groupactiononequilateraltriangle x y z z y x b

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CharacterTheoryandtheSymplecticGroup Groupactiononequilateraltriangle x y z y x z c

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CharacterTheoryandtheSymplecticGroup MultiplicationfortheGroup S 3 x y z z y x b y x z r 120 eabcr 120 r 240 e eabcr 120 r 240 a aer 120 r 240 bc b br 240 er 120 ca c cr 120 r 240 eab r 120 r 120 cabr 240 e r 240 r 240 bcaer 120

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CharacterTheoryandtheSymplecticGroup Thegroupactionasamatrix x y z z x y r 120 2 6 4 001 100 010 3 7 5 2 6 4 x y z 3 7 5 = 2 6 4 z x y 3 7 5

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CharacterTheoryandtheSymplecticGroup Representations Denition Representation :Amap X fromthegroup G tothegroupof n n invertiblematricesthatsatisesthefollowingproperty: X g 1 g 2 = X g 1 X g 2 forall g 1 ; g 2 2 G

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CharacterTheoryandtheSymplecticGroup RepresentationsandCharacters Denition Representation :Amap X fromthegroup G tothesetoflinear mapsonaspacethatsatisfythefollowingproperty: X g 1 g 2 = X g 1 X g 2 forall g 1 ; g 2 2 G Example S 3 er 120 r 240 abc X 2 6 4 100 010 001 3 7 5 2 6 4 001 100 010 3 7 5 2 6 4 010 001 100 3 7 5 2 6 4 010 100 001 3 7 5 2 6 4 001 010 100 3 7 5 2 6 4 100 001 010 3 7 5

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CharacterTheoryandtheSymplecticGroup CharactersDenition Denition A Character ofarepresentation X overanitedimensional vectorspaceisthetraceoftherepresentation X i.e.: g = Tr X g forall g 2 G

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CharacterTheoryandtheSymplecticGroup RepresentationsandCharacters Denition A Character ofarepresentation X isthetraceofthe representation X i.e.: g = Tr X g forall g 2 G Example S 3 er 120 r 240 abc X 2 6 4 100 010 001 3 7 5 2 6 4 001 100 010 3 7 5 2 6 4 010 001 100 3 7 5 2 6 4 010 100 001 3 7 5 2 6 4 001 010 100 3 7 5 2 6 4 100 001 010 3 7 5 300111

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CharacterTheoryandtheSymplecticGroup CharacterTables Agroupcanbepartitionedinto conjugacyclasses .For example,onthetopofthetablebelow,wehavepartitioned our S 3 groupelementsintoconjugacyclasses.Ineachset, eachelementisconjugatetotheotherelementsinthesame set,andnootherset. Columnsofcharactertable:conjugacyclasses Rowsofcharactertable:irreduciblecharacters Example f e gf a ; b ; c gf r 120 ; r 240 g 0 111 1 1 )]TJ/F16 10.9091 Tf 8.485 0 Td [(11 2 2 )]TJ/F16 10.9091 Tf 8.485 0 Td [(10

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CharacterTheoryandtheSymplecticGroup CharacterTables Agroupcanbepartitionedinto conjugacyclasses .For example,onthetopofthetablebelow,wehavepartitioned our S 3 groupelementsintoconjugacyclasses.Ineachset, eachelementisconjugatetotheotherelementsinthesame set,andnootherset. Columnsofcharactertable:conjugacyclasses Rowsofcharactertable:irreduciblecharacters Example f e gf a ; b ; c gf r 120 ; r 240 g 0 111 1 1 )]TJ/F16 10.9091 Tf 8.485 0 Td [(11 2 2 )]TJ/F16 10.9091 Tf 8.485 0 Td [(10

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CharacterTheoryandtheSymplecticGroup CharacterTables Agroupcanbepartitionedinto conjugacyclasses .For example,onthetopofthetablebelow,wehavepartitioned our S 3 groupelementsintoconjugacyclasses.Ineachset, eachelementisconjugatetotheotherelementsinthesame set,andnootherset. Columnsofcharactertable:conjugacyclasses Rowsofcharactertable:irreduciblecharacters Example f e gf a ; b ; c gf r 120 ; r 240 g 0 111 1 1 )]TJ/F16 10.9091 Tf 8.485 0 Td [(11 2 2 )]TJ/F16 10.9091 Tf 8.485 0 Td [(10

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CharacterTheoryandtheSymplecticGroup Theconjecture MainQuestioninCharacterTheory:Giventhecharactertable, whatcanwesayaboutthegroupitself? ConjectureRizo-SchaeerFry-Vallejo2019 Let p 2f 2 ; 3 g .Let G beanitegroupandlet B bea p -blockof G withnontrivialdefectgroup D .Thenthenumberof 1 -xed elementsof Irr 0 B is p ifandonlyif D iscyclic.

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CharacterTheoryandtheSymplecticGroup Theconjecture MainQuestioninCharacterTheory:Giventhecharactertable, whatcanwesayaboutthegroupitself? ConjectureRizo-SchaeerFry-Vallejo2019 Let p 2f 2 ; 3 g .Let G beanitegroupandlet B bea p -blockof G withnontrivialdefectgroup D .Thenthenumberof 1 -xed elementsof Irr 0 B is p ifandonlyif D iscyclic.

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CharacterTheoryandtheSymplecticGroup Blocks ConjectureRizo-SchaeerFry-Vallejo2019 Let p 2f 2 ; 3 g .Let G beanitegroupandlet B bea p -blockof G withnontrivialdefectgroup D .Thenthenumberof 1 -xed elementsof Irr 0 B is p ifandonlyif D iscyclic. Example p =2 f e gf a ; b ; c gf r 120 ; r 240 g BlocksDefectGroups 0 111 B 0 f e ; a g ; f e ; b g ; f e ; c g 1 1 )]TJ/F16 10.9091 Tf 8.485 0 Td [(11 B 0 f e ; a g ; f e ; b g ; f e ; c g 2 2 )]TJ/F16 10.9091 Tf 8.485 0 Td [(10 B 1 f e g p =3 f e gf a ; b ; c gf r 120 ; r 240 g BlocksDefectGroups 0 111 B 0 f e ; r 120 ; r 240 g 1 1 )]TJ/F16 10.9091 Tf 8.485 0 Td [(11 B 0 f e ; r 120 ; r 240 g 2 2 )]TJ/F16 10.9091 Tf 8.485 0 Td [(10 B 0 f e ; r 120 ; r 240 g

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CharacterTheoryandtheSymplecticGroup Irr 0 B ConjectureRizo-SchaeerFry-Vallejo2019 Let p 2f 2 ; 3 g .Let G beanitegroupandlet B bea p -blockof G withnontrivialdefectgroup D .Thenthenumberof 1 -xed elementsof Irr 0 B is p ifandonlyif D iscyclic. Irr 0 B isasubsetoftheirreduciblecharactersofagroupina givenblock B ,determinedbytheirvalueattheidentityelement, thesizeofthedefectgroupandtheprimeintheconjecture. Example p =2 f e gf a ; b ; c gf r 120 ; r 240 g BlocksDefectGroups 0 111 B 0 f e ; a g ; f e ; b g ; f e ; c g 1 1 )]TJ/F16 10.9091 Tf 8.485 0 Td [(11 B 0 f e ; a g ; f e ; b g ; f e ; c g 2 2 )]TJ/F16 10.9091 Tf 8.485 0 Td [(10 B 1 f e g

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CharacterTheoryandtheSymplecticGroup 1 -xed ConjectureRizo-SchaeerFry-Vallejo2019 Let p 2f 2 ; 3 g .Let G beanitegroupandlet B bea p -blockof G withnontrivialdefectgroup D .Thenthenumberof 1 -xed elementsof Irr 0 B is p ifandonlyif D iscyclic. 1 isa Galoisautomorphism ,atypeofgroupactionthatactson charactersdependingontheothervalues. Example p =2 f e gf a ; b ; c gf r 120 ; r 240 g BlocksDefectGroups 0 111 B 0 f e ; a g ; f e ; b g ; f e ; c g 1 1 )]TJ/F16 10.9091 Tf 8.485 0 Td [(11 B 0 f e ; a g ; f e ; b g ; f e ; c g 2 2 )]TJ/F16 10.9091 Tf 8.485 0 Td [(10 B 1 f e g

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CharacterTheoryandtheSymplecticGroup TheSymplecticGroup Sp 4 q ConjectureRizo-SchaeerFry-Vallejo2019 Let p 2f 2 ; 3 g .Let G beanitegroupandlet B bea p -blockof G withnontrivialdefectgroup D .Thenthenumberof 1 -xed elementsof Irr 0 B is p ifandonlyif D iscyclic. Ournitegroup: Denition Sp 4 q isthegroupof4 4matrices M withentriesinthenite eld F q ,where q isapowerofsomeprime,suchthat M satises thefollowing: MJM T = J where J is 2 6 6 4 0100 )]TJ/F16 10.9091 Tf 8.484 0 Td [(1000 0001 00 )]TJ/F16 10.9091 Tf 8.485 0 Td [(10 3 7 7 5

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CharacterTheoryandtheSymplecticGroup TheSymplecticGroup Sp 4 q ConjectureRizo-SchaeerFry-Vallejo2019 Let p 2f 2 ; 3 g .Let G beanitegroupandlet B bea p -blockof G withnontrivialdefectgroup D .Thenthenumberof 1 -xed elementsof Irr 0 B is p ifandonlyif D iscyclic. Ournitegroup: Denition Sp 4 q isthegroupof4 4matrices M withentriesinthenite eld F q ,where q isapowerofsomeprime,suchthat M satises thefollowing: MJM T = J where J is 2 6 6 4 0100 )]TJ/F16 10.9091 Tf 8.484 0 Td [(1000 0001 00 )]TJ/F16 10.9091 Tf 8.485 0 Td [(10 3 7 7 5

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CharacterTheoryandtheSymplecticGroup TheSymplecticGroup Sp 4 q ConjectureRizo-SchaeerFry-Vallejo2019 Let p 2f 2 ; 3 g .Let G beanitegroupandlet B bea p -blockof G withnontrivialdefectgroup D .Thenthenumberof 1 -xed elementsof Irr 0 B is p ifandonlyif D iscyclic. Ournitegroup: Denition Sp 4 q isthegroupof4 4matrices M withentriesinthenite eld F q ,where q isapowerofsomeprime,suchthat M satises thefollowing: MJM T = J where J is 2 6 6 4 0100 )]TJ/F16 10.9091 Tf 8.484 0 Td [(1000 0001 00 )]TJ/F16 10.9091 Tf 8.485 0 Td [(10 3 7 7 5

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CharacterTheoryandtheSymplecticGroup OurResults! TheoremFleming-Ruiz-Tusa-Vigil Let p 2f 2 ; 3 g .Let G = Sp 4 q with q apowerofaprime dierentthan p .Let B bea p -blockof G withnontrivialdefect group D .Thenthenumberof 1 -xedelementsof Irr 0 B is p if andonlyif D iscyclic. p =2 q apowerofanoddprime GabrielVigil p =3 q apower2 CourtneyFleming/AlexTusa p =3 q apowerofanoddprime 6 =3 JosephDayngerRuiz

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CharacterTheoryandtheSymplecticGroup Thanks! ThankstoDr.MandiandNSFgrantnumberDMS-1801156 Andthanksforlistening!

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CharacterTheoryandtheSymplecticGroup ThanksforListening! Ifyouhaveanyquestions,pleaseemailusat CharacterTheoryAndTheSymplecticGroupQuestions@outlook.com