ThePrimitiveDistance-TransitiveRepresentationsoftheFischerGroups
StephenA.Linton,KlausLuxandLeonardH.Soicher
CONTENTS
1.Introduction
2.OrbitalGraphs,Distance-TransitiveRepresentations,andtheMainTheorem3.TheGeneralApproach
4.TheComputationalStudyofPermutationCharacters5.TheMultiplicity-FreePrimitivePermutationRepresentationsoftheFischerGroups
6.AnalysisoftheMultiplicity-FreePrimitiveRepresentationsofRankGreaterThan3
Weclassifytheprimitivedistance-transitiverepresentationsoftheFischersporadicsimplegroupsandtheirautomorphismgroups.Itturnsoutthattheonlyprimitivedistance-transitiverepresentationsofthesegroupsaretheirrank3representations.Intheprocessofourwork,wealsoclassifyandstudytheprimitivemultiplicity-freepermutationrepresentationsoftheseFischergroups.Ourmethods,whichwedescribeinsomedetail,demonstratetheuseofcomputationalandrandomizedtechniquesintheclassificationofdistance-transitivegraphsandthestudyofverylargepermutationrepresentations.
1.INTRODUCTION
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2.ORBITALGRAPHS,DISTANCE-TRANSITIVE
REPRESENTATIONS,ANDTHEMAINTHEOREM
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Theorem2.1.
1.
2.
3.
4.
5.
3.THEGENERALAPPROACH
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Remark.
4.THECOMPUTATIONALSTUDYOFPERMUTATION
CHARACTERSDeterminingaPermutationCharacter
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UsefulTrickstoShowThatCertainCharactersAreNotMultiplicity-Free
Lemma4.1.
Lemma4.2.
OnthePermutationCharactersofG2
Lemma4.3.
Linton,LuxandSoicher:ThePrimitiveDistance-TransitiveRepresentationsoftheFischerGroups241
5.THEMULTIPLICITY-FREEPRIMITIVEPERMUTATION
REPRESENTATIONSOFTHEFISCHERGROUPS
Theorem5.1.
1.2.3.4.5.6.7.
Theorem5.2.
1.
2.3.
4.
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Theorem5.3.
1.2.
3.
Theorem5.4.
1.2.3.4.
5.
Proof.
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Theorem5.5.
1.
2.
3.
Proof.
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Fi222onO82S3
2
Theorem6.1.
6.ANALYSISOFTHEMULTIPLICITY-FREEPRIMITIVE
REPRESENTATIONSOFRANKGREATERTHAN3
Proof.
Fi222on210M22Theorem6.2.
Proof.
2
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Proof.
Remark.
Fi222on26
S62
Theorem6.3.
2
Fi222on2F42
Theorem6.4.
2
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Proof.
Fi23onS82
Theorem6.5.
Proof.
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Fi
on
Theorem6.6.
Proof.
Fi242onO102
Theorem6.7.
2
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Proof.
TABLE1.
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Fi242on37O73
2
Theorem6.8.
Proof.
TABLE2.
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REFERENCES
Linton,LuxandSoicher:ThePrimitiveDistance-TransitiveRepresentationsoftheFischerGroups253
