Explorations of the Aldous Order on Representations of the Symmetric Group

57 Pages·2012·0.47 MB·English

by  Jackson Newhouse

Explorations of the Aldous Order on
Representations of the Symmetric Group
Jackson Newhouse
Matt Davis, Advisor
Michael Orrison, Reader
May,2012
DepartmentofMathematics
Copyright(cid:13)c 2012JacksonNewhouse.
TheauthorgrantsHarveyMuddCollegeandtheClaremontCollegesLibrarythe
nonexclusive right to make this work available for noncommercial, educational
purposes,providedthatthiscopyrightstatementappearsonthereproducedma-
terialsandnoticeisgiventhatthecopyingisbypermissionoftheauthor. Todis-
seminateotherwiseortorepublishrequireswrittenpermissionfromtheauthor.
Abstract
TheAldousorderisanorderingofrepresentationsofthesymmetricgroup
motivated by the Aldous conjecture, a conjecture about random processes
proved in 2009. In general, the Aldous order is very difficult to compute,
and the proper relations have yet to be determined even for small cases.
However, by restricting the problem down to Young–Jucys–Murphy ele-
ments, the problem becomes explicitly combinatorial. This approach has
led to many novel insights, whose proofs are simple and elegant. How-
ever, there remain many open questions related to the Aldous order, both
ingeneralandfortheYoung–Jucys–Murphyelements.

Contents
Abstract iii
Acknowledgments vii
I BackgroundMaterial 1
1 RepresentationTheoryofFiniteGroups 3
1.1 TheGroupAlgebra . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 RepresentationsofaFiniteGroup . . . . . . . . . . . . . . . . 5
2 RepresentationTheoryoftheSymmetricGroups 7
2.1 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 YoungDiagramsandStandardYoungTableaux . . . . . . . 9
2.3 StandardYoungTableaux . . . . . . . . . . . . . . . . . . . . 9
2.4 TheIrreducibleRepresentationsofS . . . . . . . . . . . . . 11
n
2.5 TheTrivialandAlternatingRepresentations . . . . . . . . . . 12
2.6 LexicographicandDominanceOrders . . . . . . . . . . . . . 12
2.7 YoungModules . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.8 DecompositionofYoungModules . . . . . . . . . . . . . . . 14
II GeneralAldousOrder 17
3 Randomσ-InterchangeProcesses 19
3.1 TheInterchangeProcess . . . . . . . . . . . . . . . . . . . . . 19
3.2 IntensityMatricesandYoungModules . . . . . . . . . . . . . 20
3.3 Decompositionof Mσ . . . . . . . . . . . . . . . . . . . . . . . 22
3.4 TheUniformDistribution . . . . . . . . . . . . . . . . . . . . 22
vi Contents
4 DefinitionsandSuperordersoftheAldousOrder 25
4.1 TheAldousConjecture . . . . . . . . . . . . . . . . . . . . . . 25
4.2 TheAldousOrderDefinition . . . . . . . . . . . . . . . . . . 26
4.3 PreviousWork . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.4 AlternateFormulations . . . . . . . . . . . . . . . . . . . . . . 27
4.5 Propertiesoftheρ-Norm . . . . . . . . . . . . . . . . . . . . . 30
4.6 SuperordersoftheAldousOrder . . . . . . . . . . . . . . . . 31
III TheYJMAldousOrder 33
5 BasicPropertiesoftheYJMAldousOrder 35
5.1 ComputingtheYJMOrder . . . . . . . . . . . . . . . . . . . . 35
5.2 FrontandBackSums . . . . . . . . . . . . . . . . . . . . . . . 37
6 TwoTheoremsintheYJMAldousOrder 41
6.1 FirstRowIncomparability . . . . . . . . . . . . . . . . . . . . 41
6.2 StrictDominance . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.3 ConjecturesandFutureWork . . . . . . . . . . . . . . . . . . 47
Bibliography 49
Acknowledgments
I would like to thank Professor Matt Davis for his patience and guidance
as my advisor. I owe a particular debt to the special topics course he and
Professor Michael Orrison taught in the spring of 2011 on Representation
Theory of the Symmetric Group. It was one of the most enjoyable and
challengingcoursesIhavetakenatHarveyMuddCollege.

Part I
Background Material