Table Of ContentTexts and Readings in Mathematics 53
V Lakshmibai
Justin Brown
Flag Varieties
An Interplay of Geometry, Combinatorics,
and Representation Theory
Second Edition
Texts and Readings in Mathematics
Volume 53
Advisory Editor
C. S. Seshadri, Chennai Mathematical Institute, Chennai
Managing Editor
Rajendra Bhatia, Indian Statistical Institute, New Delhi
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V. Balaji, Chennai Mathematical Institute, Chennai
R. B. Bapat, Indian Statistical Institute, New Delhi
V. S. Borkar, Indian Institute of Technology, Mumbai
T. R. Ramadas, Chennai Mathematical Institute, Chennai
V. Srinivas, Tata Institute of Fundamental Research, Mumbai
Technical Editor
P. Vanchinathan, Vellore Institute of Technology, Chennai
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V. Lakshmibai Justin Brown
(cid:129)
Flag Varieties
An Interplay of Geometry, Combinatorics,
and Representation Theory
Second Edition
123
V.Lakshmibai JustinBrown
Northeastern University Olivet NazareneUniversity
Boston, MA, USA Bourbonnais, IL, USA
ISSN 2366-8725 (electronic)
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Preface
This book provides an introduction to (cid:13)ag varieties and their Schubert subva-
rieties. The book portrays (cid:13)ag varieties as an interplay of algebraic geometry,
algebraic groups, combinatorics, and representation theory.
Afterdiscussingtherepresentationtheoryof(cid:12)nitegroups,thepolynomial
representations (in characteristic zero) of the general linear group are obtained
by relating the representation theory of the general linear group to that of the
symmetricgroup(usingSchur-Weylduality).SincetheLiealgebraofasemisim-
ple algebraic group plays a crucial role in the structure theory of semisimple
algebraic groups, the book discusses the structure theory and the representa-
tion theory of complex semisimple Lie algebras. Since Bruhat decomposition is
at the heart of the study of the (cid:13)ag variety, the book gives a quick treatment
of the generalities on algebraic groups leading to the root system and Bruhat
decomposition in reductive algebraic groups.
The nucleus of this book is the geometry of the Grassmannian and (cid:13)ag
varieties. A knowledge of Grassmannian and (cid:13)ag varieties is indispensable for
anyprospectivegraduatestudentworkingintheareaofalgebraicgeometry.We
hopethatthisbookwillserveasareferenceforbasicresultsonGrassmannian,
(cid:13)ag, and Schubert varieties as well as the relationship between the geometric
aspectsofthesevarietiesandtherepresentationtheoryofsemisimplealgebraic
groups.
The prerequisite for this book is some familiarity with commutative alge-
bra,algebraicgeometry,andalgebraicgroups.Abasicreferencetocommutative
algebra is [17], algebraic geometry [28], and algebraic groups [5, 36]. The basic
results from commutative algebra and algebraic geometry are summarized in
Chapter 1. We have mostly used standard notation and terminology and have
triedtokeepnotationtoaminimum.Throughoutthebook,wehavenumbered
theorems, lemmas, propositions etc., in order according to their chapter and
section;forexample,3.2.4referstothefourthitemofthesecondsectioninthe
third chapter.
This book can be used for an introductory course on (cid:13)ag varieties. The
materialcoveredinthisbookshouldprovideadequatepreparationforgraduate
studentsandresearchersintheareaofalgebraicgeometryandalgebraicgroups.
Fortheinterestedreader,wehaveincludedseveralexercisesattheendofalmost
v
vi Preface
every chapter; most of these exercises can also be found in the standard texts
on their respective subjects.
Acknowledgments:V.LakshmibaithankstheorganizersK.Uhlenbeckand
C. Terng of the Institute for Advanced Study Program for Women and Mathe-
maticson\AlgebraicGeometryandGroupActions,"May2007,forinvitingher
to give lectures on \Flag Variety," and also the Institute for Advanced Study
for the hospitality extended to her during her stay there.
J.BrownthanksallofthosewhoparticipatedintheFlagVarietiescourse
at Northeastern University, speci(cid:12)cally K. Webster and M. Fries for their con-
tributions and suggestions. He also thanks his wife, Jody, for her love and
support.
September 1, 2007 V. Lakshmibai
Boston, MA, USA J. Brown
Preface to the Second Edition
In this second edition, we have added two recent results (from [42] and
[32], respectively) on Schubert varieties in the Grassmannian. The (cid:12)rst result,
whichhasbeenaddedasChapter15,givesafreeresolutionofcertainSchubert
singularities. The second result, which has been added as Chapter 16, is about
certain Levi subgroup actions on Schubert varieties in the Grassmannian and
derivessomeinterestinggeometricandrepresentation-theoreticconsequences.
Contents
Preface v
Introduction xiii
1 Preliminaries 1
1.1 Commutative Algebra . . . . . . . . . . . . . . . . . . . . . . 1
1.2 A(cid:14)ne Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Projective Varieties . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Schemes - A(cid:14)ne and Projective . . . . . . . . . . . . . . . . 10
1.5 The Scheme Spec(A). . . . . . . . . . . . . . . . . . . . . . . 12
1.6 The Scheme Proj(S) . . . . . . . . . . . . . . . . . . . . . . . 13
1.7 Sheaves of O -Modules . . . . . . . . . . . . . . . . . . . . . 14
X
1.8 Attributes of Varieties . . . . . . . . . . . . . . . . . . . . . . 18
2 Structure Theory of Semisimple Rings 21
2.1 Semisimple Modules . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Semisimple Rings . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Brauer Groups and Central Simple Algebras . . . . . . . . . 32
2.4 The Group Algebra, K[G] . . . . . . . . . . . . . . . . . . . . 34
2.5 The Center of K[G] . . . . . . . . . . . . . . . . . . . . . . . 35
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 Representation Theory of Finite Groups 39
3.1 Representations of G . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Characters of Representations . . . . . . . . . . . . . . . . . 40
3.3 Ordinary Representations . . . . . . . . . . . . . . . . . . . . 43
3.4 Tensor Product of Representations . . . . . . . . . . . . . . . 44
3.5 Contragradient Representations . . . . . . . . . . . . . . . . 45
3.6 Restrictions and Inductions . . . . . . . . . . . . . . . . . . . 45
3.7 Character Group of G . . . . . . . . . . . . . . . . . . . . . . 47
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
vii
viii Contents
4 Representation Theory of the Symmetric Group 49
4.1 The Symmetric Group S . . . . . . . . . . . . . . . . . . . . 49
n
4.2 Frobenius-Young Modules . . . . . . . . . . . . . . . . . . . . 54
4.3 Specht Modules . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.5 Proof of Theorem 4.3.7 . . . . . . . . . . . . . . . . . . . . . 63
4.6 Representation Theory of A . . . . . . . . . . . . . . . . . . 65
n
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5 Symmetric Polynomials 69
5.1 Notation and Motivation . . . . . . . . . . . . . . . . . . . . 69
5.2 Several Bases for S(n) . . . . . . . . . . . . . . . . . . . . . 70
5.3 Kostka Numbers & Determinantal Formulas . . . . . . . . . 71
5.4 Results on (cid:31) . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
(cid:21)
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6 Schur-Weyl Duality and the Relationship Between Representations of
S and GL (C) 79
d n
6.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.2 Schur-Weyl Duality . . . . . . . . . . . . . . . . . . . . . . . 81
6.3 Characters of the Schur Modules . . . . . . . . . . . . . . . . 83
6.4 Schur Module Representations of SL (C) . . . . . . . . . . . 84
n
6.5 Representations of GL (C) . . . . . . . . . . . . . . . . . . . 85
n
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7 Structure Theory of Complex Semisimple Lie Algebras 89
7.1 Introduction to Semisimple Lie Algebras. . . . . . . . . . . . 89
7.2 The Exponential Map in Characteristic Zero . . . . . . . . . 91
7.3 Structure of Semisimple Lie Algebras . . . . . . . . . . . . . 91
7.4 Jordan Decomposition in Semisimple Lie Algebras . . . . . . 95
7.5 The Lie Algebra sl (C) . . . . . . . . . . . . . . . . . . . . . 96
n
7.6 Cartan Subalgebras . . . . . . . . . . . . . . . . . . . . . . . 97
7.7 Root Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.8 Structure Theory of sl (C) . . . . . . . . . . . . . . . . . . . 100
n
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
8 Representation Theory of Complex Semisimple Lie Algebras 103
8.1 Representations of g . . . . . . . . . . . . . . . . . . . . . . . 103
8.2 Weight Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 107
8.3 Finite Dimensional Modules . . . . . . . . . . . . . . . . . . 109
8.4 Fundamental Weights . . . . . . . . . . . . . . . . . . . . . . 112
8.5 Dimension and Character Formulas . . . . . . . . . . . . . . 113
8.6 Irreducible sl (C)-Modules . . . . . . . . . . . . . . . . . . . 113
n
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Contents ix
9 Generalities on Algebraic Groups 115
9.1 Algebraic Groups and Their Lie Algebras . . . . . . . . . . . 115
9.2 The Tangent Space . . . . . . . . . . . . . . . . . . . . . . . 116
9.3 Jordan Decomposition in G . . . . . . . . . . . . . . . . . . . 117
9.4 Variety Structure on G=H . . . . . . . . . . . . . . . . . . . 121
9.5 The Flag Variety . . . . . . . . . . . . . . . . . . . . . . . . . 125
9.6 Structure of Connected Solvable Groups . . . . . . . . . . . . 126
9.7 Borel Fixed Point Theorem . . . . . . . . . . . . . . . . . . . 128
9.8 Variety of Borel Subgroups . . . . . . . . . . . . . . . . . . . 132
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
10 Structure Theory of Reductive Groups 135
10.1 Cartan Subgroups . . . . . . . . . . . . . . . . . . . . . . . . 135
10.2 The Weyl Group . . . . . . . . . . . . . . . . . . . . . . . . . 136
10.3 Regular and Singular Tori . . . . . . . . . . . . . . . . . . . . 137
10.4 Semisimple Rank 1. . . . . . . . . . . . . . . . . . . . . . . . 139
10.5 One Parameter Subgroups . . . . . . . . . . . . . . . . . . . 140
10.6 Reductive Groups . . . . . . . . . . . . . . . . . . . . . . . . 143
10.7 Almost Simple Groups. . . . . . . . . . . . . . . . . . . . . . 146
10.8 Schubert Varieties & Bruhat Decomposition . . . . . . . . . 147
10.9 Standard Parabolic Subgroups . . . . . . . . . . . . . . . . . 150
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
11 Representation Theory of Semisimple Algebraic Groups 153
11.1 Weight Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 153
11.2 Geometric Realization of V((cid:21)) . . . . . . . . . . . . . . . . . 157
12 Geometry of the Grassmannian, Flag and their Schubert Varieties via
Standard Monomial Theory 165
12.1 Grassmannian Variety . . . . . . . . . . . . . . . . . . . . . . 165
12.2 G Identi(cid:12)ed with a Homogeneous Space . . . . . . . . . . 168
d;n
12.3 Schubert Varieties . . . . . . . . . . . . . . . . . . . . . . . . 169
12.4 Standard Monomials . . . . . . . . . . . . . . . . . . . . . . . 171
12.5 Equations De(cid:12)ning Schubert Varieties . . . . . . . . . . . . . 173
12.6 Unions of Schubert Varieties . . . . . . . . . . . . . . . . . . 174
12.7 Vanishing Theorems . . . . . . . . . . . . . . . . . . . . . . . 176
12.8 Results for F . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
13 Singular Locus of a Schubert Variety in the Flag Variety SL =B 187
n
13.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
13.2 Singular Loci of Schubert Varieties . . . . . . . . . . . . . . . 188
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
