Representations of quantum algebras and combinatorics of Young tableaux

169 Pages·2002·1.074 MB·English

by  Susumu Ariki

University
L
ECTURE
Series
Volume 26
Representations
of Quantum Algebras
and Combinatorics
of Young Tableaux
Susumu Ariki
American Mathematical Society
Representations
of Quantum Algebras
and Combinatorics
of Young Tableaux

University
L
ECTURE
Series
Volume 26
Representations
of Quantum Algebras
and Combinatorics
of Young Tableaux
Susumu Ariki
ERMIACAN MΑΓΕΩΜΕAΤTΡHΗEΤΟMΣ AΜTΗICΕΙΣΙΤΩALSOYCTIE
FOUNDED 1888
American Mathematical Society
Providence, Rhode Island
EDITORIAL COMMITTEE
Jerry L. Bona (Chair) Nigel J. Hitchin
Jean-Luc Brylinski Nicolai Reshetikhin
A(1) -GATA KEIRYOSHIGUN NO HYOGENRON TO KUMIAWASERON
r−1
(REPRESENTATION OF QUANTUM ALGEBRAS OF TYPE A(1)
r−1
AND COMBINATORICS OF YOUNG TABLEAUX)
by Susumu Ariki
Originally published in Japanese
by Sophia University, Tokyo, 2000
Translated from the Japanese and revised by the author
2000 Mathematics Subject Classification. Primary 05E10, 17B37, 17B67, 20C08;
Secondary 14M15, 16D90, 16G20, 20C33.
Library of Congress Cataloging-in-Publication Data
Ariki,Susumu,1959–
Representations of quantum algebras and combinatorics of Young tableaux / Susumu Ariki ;
[translatedfromtheJapaneseandrevisedbytheauthor].
p.cm. —(Universitylectureseries,ISSN1047-3998;v.26)
Includesbibliographicalreferencesandindex.
ISBN0-8218-3232-8(acid-freepaper)
1. Quantum groups. 2. Representations of groups. 3. Bases (Linear topological spaces)
4.Youngtableaux. I.Title. II.Universitylectureseries(Providence,R.I.);26.
QA176.A7513 2002
530.14(cid:2)3—dc21 2002025869
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Contents
Preface vii
Chapter 1. Introduction 1
1.1. How do you do ? 1
1.2. What are we interested in ? 1
1.3. Enveloping algebras 3
Chapter 2. The Serre relations 9
2.1. The Serre relations 9
2.2. The quantum algebra of type Ar−1 13
Chapter 3. Kac-Moody Lie algebras 15
3.1. Lie algebras by generators and relations 15
3.2. Kac-Moody Lie algebras 17
3.3. The quantum algebra of type A(1) 21
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Chapter 4. Crystal bases of U -modules 23
v
4.1. Integrable modules 23
4.2. The Kashiwara operators 24
4.3. Crystal bases 27
Chapter 5. The tensor product of crystals 29
5.1. Basics of crystal bases 29
5.2. Tensor products of crystal bases 34
Chapter 6. Crystal bases of U− 37
v
6.1. Triangular decomposition of U 37
v
6.2. Integrable highest weight modules 39
6.3. Crystal bases of U− 42
v
Chapter 7. The canonical basis 47
7.1. A review of Lusztig’s canonical basis 47
7.2. The lattice of the canonical basis 51
Chapter 8. Existence and uniqueness (part I) 55
8.1. Preparatory lemmas 55
8.2. The first main theorem 59
Chapter 9. Existence and uniqueness (part II) 61
9.1. Preparatory results 61
9.2. The second main theorem 65
v
vi CONTENTS
Chapter 10. The Hayashi realization 75
10.1. Partitions and the Hayashi realization 75
10.2. Generalization to the case of multipartitions 85
Chapter 11. Description of the crystal graph of V(Λ) 87
11.1. A theorem for proving the Misra-Miwa theorem 87
11.2. The Misra-Miwa theorem 94
Chapter 12. An overview of the applications to Hecke algebras 97
12.1. The Hecke algebra of type G(m,1,n) 97
12.2. Consequences of Theorem 12.5 101
Chapter 13. The Hecke algebra of type G(m,1,n) 105
13.1. The affine Hecke algebra 105
13.2. Semi-normal representations 107
13.3. The decomposition map 111
13.4. Specht module theory 113
13.5. A theorem of Morita type 117
13.6. Refined induction and restriction functors 118
Chapter 14. The proof of Theorem 12.5 123
14.1. Representations of a cyclic quiver 123
14.2. The Hall algebra and the quantum algebra 127
14.3. Some results from the geometric theory 133
14.4. Proof of the generalized LLT conjecture 142
Chapter 15. Reference guide 147
Bibliography 149
Index 157
Preface
Quantum groups are in fact not groups. They are also called quantized en-
veloping algebras, or quantum algebras for short. They were bornin mathematical
physicsandhaveevolvedintoavastareaofresearch. Inparticular,theyhavefound
applicationsinalgebraicgroupsandgivenrisetobigprogressintheLusztigconjec-
ture for algebraic groups. But I have another story totell: these quantum algebras
also gave rise to new combinatorial objects and have influenced the combinatorics
related to representation theories. This research area is called “combinatorial rep-
resentation theory”.
These lecture notes are based on my lectures delivered at Sophia university in
1997, and are intended for graduate students who have interests in this area.
In the preparation of the lectures, I benefitted from two important papers,
[Kashiwara] and [Lusztig]. In fact, my primary intention was to introduce the
reader to the theory of crystal bases and canonical bases by working out special
examples, quantum algebras of type A(1) .
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In the lectures, I have named fundamental theorems about crystal bases and
canonical bases of quantum algebras as the first, the second and the third main
theorems of Kashiwara and Lusztig. I hope that the naming is accepted by the
society of mathematicians.
Theplanofthebookisasfollows. Thefirstthreechaptersareapreparationto
start running. In the 4th to the 6th chapters, we establish basic notions of crystal
bases. We then introduce canonical bases in Chapter 7 and prove fundamental
theorems in the subsequent two chapters. These chapters have flavors of the gen-
eral theory, although we are content with our examples, the quantum algebras of
type A(1) . In the next two chapters, we turn to combinatorics. We prove the
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combinatorial construction of crystal bases of Fock spaces due to Misra and Miwa.
In the 12th chapter, we summarize its applications to the representation theory of
cyclotomic Hecke algebras. The 13th and 14th chapters are devoted to the proof
of my main theorem stated in Chapter 12. The final chapter is a guide for further
reading. Thelistisnotintendedtobecompleteofcourse, andreflectsmypersonal
research interests.
Ifthereaderhassomefamiliaritywithrepresentationtheory,Irecommendskip-
pingthefirstthreechapters. Ifhe/shehassomespecialtyinthisfield,Irecommend
starting with the 7th chapter.
IwouldliketothankProfessorBhamaSrinivasanandAndrewMathasforread-
ing the manuscript, and my wife Tomoko for many things. During the preparation
ofthesenotes,IwaspartiallysupportedbytheJSPS-DFGJapanese-Germancoop-
erative science program “Representation Theory of Finite and Algebraic Groups”.
Susumu Ariki
vii

CHAPTER 1
Introduction
1.1. How do you do ?
These lectures are for graduate students who know the basics of the represen-
tation theory of finite groups and artinian rings. Through these lectures, you will
be exposed to some recent research in mathematics. Since we have to skip the
proofs of several theorems at several points to keep these lectures elementary, you
are encouraged to read the original papers for these parts in the second reading.
In the first reading, I recommend trusting the results so as to make your life easy.
Although I skip proofs at several points, these notes are basically written in the
“theorem and proof ”style.
ThemainexampleweuseisthequantumalgebraoftypeA(1) . Thepurposeof
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thefirst halfofthisbookistoexplainthegeneraltheoryofcrystalbasesusingthis
example. Inthesecondhalf, weexplainseveralinterestingresultsusingYoungdia-
grams. I hope that the reader finds it interesting to do research in “Combinatorial
Representation Theory”after reading these notes.
1.2. What are we interested in ?
Whenyoustartyourprofessionaleducationinmathematics,yousoonencounter
the notion of a group. It is a mathematical device used to describe symmetries in
nature. Itisanideathatdevelopedconcurrentlyinseveralareas,suchasgeometry,
number theory and the algebra of polynomial equations, with an axiomatic defini-
tion coming in the middle of the 19th century. As you already know, the Galois
theoryisthemostfamousapplicationofthegrouptheory. Inmoderntimesgroups
are widely used in many areas and they have little to do with equations. They are
also used in essential ways in physics and chemistry. For example, the groups used
in gauge theories and classification of elementary particles are called Lie groups.
A typical example of a Lie group is the matrix group
(cid:2) (cid:3)
GL(n,C)= X ∈M(n,n,C)|det(X)(cid:3)=0 ,
but there are other examples as well. Let us consider a group G. Often G will
be described as a group of matrices; there are many different ways of doing this,
usually using matrices of different sizes, although the group behind them all is the
same. Namely,therearemanywaystoassociateamatrixρ(X)withX ∈Ginsuch
awaythattheproductoftwoelementscorrespondstotheproductoftheassociated
matrices (i.e.ρ(X)ρ(Y)=ρ(XY)). These ρ are called representations of G. Lie
himself worked with “infinitesimal groups”, understanding that for many things it
was usually enough to consider the Lie algebras. In today’s language, we can say
thatanessential featureofLie’sworkishisdiscoverythatbylookingatthesecond
1