Linear Algebraic Monoids (eBook)

Preface 6
Contents 8
1 Introduction 12
2 Background 16
2.1 Algebraic Geometry 16
2.2 Algebraic Groups 23
2.3 Semigroups 37
2.4 Exercises 42
3 Algebraic Monoids 43
3.1 Linear Algebraic Monoids 43
3.2 Normal Monoids 47
3.3 D-monoids 48
3.4 Solvable Monoids 50
3.5 Excercises 50
4 Regularity Conditions 54
4.1 Reductive Monoids 54
4.2 Semigroup Structure of Reductive Monoids 56
4.3 Solvable Regular Monoids 57
4.4 Regular Algebraic Monoids 59
4.5 Regularity in Codimension One 61
4.6 Exercises 63
5 Classification of Reductive Monoids 65
5.1 The Extension Principle 65
5.2 Vinberg’s Approach 69
5.3 Algebraic Monoids as Spherical Varieties 72
6 Universal Constructions 80
6.1 Quotients 80
6.2 Class Groups of Reductive Monoids 82
6.3 Flat Monoids 85
6.4 Multilined Closure 93
6.5 Normalization and Representations 96
6.6 Exercises 97
7 Orbit Structure of Reductive Monoids 98
7.1 The System of Idempotents and the Type Map 98
7.2 The Cross Section Lattice and the Weyl Chamber 100
7.3 J-irreducible Monoids 103
7.4 Explicit Calculations of the Type Map 106
7.5 2-reducible Reductive Monoids 114
7.6 Type Maps in General 129
7.7 Exercises 131
8 The Analogue of the Bruhat Decomposition 133
8.1 The Renner Monoid R 133
8.2 The Analogue of the Tits System 135
8.3 Row Reduced Echelon Form 138
8.4 The Length Function on R 141
8.5 Order-Preserving Elements of R 143
8.6 The Adherence Order on R 146
8.7 The j-order, R+ and Pennell’s Theorem 149
8.8 The Adherence Order on Mn(K) 153
8.9 Exercises 156
9 Representations and Blocks of Algebraic Monoids 158
9.1 Conjugacy Classes and Adjoint Quotient 158
9.2 Rep(M) according to Doty 161
9.3 The Blocks of Mn(K) when char(K) = p > 0
9.4 The Blocks of Solvable Algebraic Monoids 165
10 Monoids of Lie Type 172
10.1 Finite Groups of Lie Type 172
10.2 Endomorphisms of Linear Algebraic Monoids 173
10.3 A Detailed Example 174
10.4 Abstract Monoids of Lie Type 177
10.5 Modular Representations of Finite Reductive Monoids 180
10.6 Exercises 189
11 Cellular Decomposition of Algebraic Monoids 191
11.1 Monoid Cells 192
11.2 Exercises 196
12 Conjugacy Classes 198
12.1 The Basic Conjugacy Theorem 198
12.2 Some Refinements 200
12.3 Putcha’s Decomposition and the Nilpotent Variety 202
13 The Centralizer of a Semisimple Element 208
13.1 Introduction 208
13.2 Main Results 209
13.3 The Structure of Rs and Ms 211
13.4 Examples 212
14 Combinatorics Related to Algebraic Monoids 216
14.1 The Adherence Order on WeW 216
14.2 Shellability and Stanley-Reisner Rings 220
14.3 Distribution of Products in Finite Monoids 220
14.4 Exercises 229
15 Survey of Related Developments 230
15.1 Complex Representation of Finite Reductive Monoids 230
15.2 Finite Semigroups and Highest Weight Categories 231
15.3 Singularities of G-embeddings 233
15.4 Cohomology of G-embeddings 234
15.5 Horospherical Varieties 235
15.6 Monoids associated with Kac-Moody Groups 235
References 237
Index 244