Introduction to the Theory of Standard Monomials: Second Edition (Texts and Readings in Mathematics Book 46)

The book is a reproduction of a course of lectures delivered by the author in 1983-84 which appeared in the Brandeis Lecture Notes series. The aim of this course was to give an introduction to the series of papers by concentrating on the case of the full linear group. In recent years, there has been great progress in standard monomial theory due to the work of Peter Littelmann. The author's lectures (reproduced in this book) remain an excellent introduction to standard monomial theory. d-origin: initial; background-clip: initial; background-position: initial; background-repeat: initial;">Standard monomial theory deals with the construction of nice bases of finite dimensional irreducible representations of semi-simple algebraic groups or, in geometric terms, nice bases of coordinate rings of flag varieties (and their Schubert subvarieties) associated with these groups. Besides its intrinsic interest, standard monomial theory has applications to the study of the geometry of Schubert varieties. Standard monomial theory has its origin in the work of Hodge, giving bases of the coordinate rings of the Grassmannian and its Schubert subvarieties by "standard monomials". In its modern form, standard monomial theory was developed by the author in a series of papers written in collaboration with V. Lakshmibai and C. Musili. In the second edition of the book, conjectures of a standard monomial theory for a general semi-simple (simply-connected) algebraic group, due to Lakshmibai, have been added as an appendix, and the bibliography has been revised Contents 7 Preface to the second edition 9 Preface to the first edition 10 Introduction 11 About the Author 13 1 Schubert Varieties in the Grassmannian 14 1.1 Plücker coordinates 14 1.2 Schubert varieties 18 1.3 Standard monomials 25 1.4 Some Applications 32 1.5 Degeneration of Schubert varieties 45 2 Standard monomial theory on SLn(k)/Q 67 2.1 Some facts about G/Q 67 2.2 Young diagrams and standard monomials 71 2.3 Linear independence of standard monomials 73 2.4 Some facts about the partial order on W/WQi 75 2.5 Preparation for the main theorem 77 2.6 Main theorem 82 2.7 Another proof for generation by standard monomials 86 3 Applications 93 3.1 Singularities of Schubert varieties 93 3.2 Vanishing theorem 97 3.3 Character formula 101 3.4 Ideal theory of Schubert varieties 105 3.5 The variety of complexes 109 4 Schubert varieties in G/Q 118 4.1 Some remarks on linear algebraic groups 118 4.2 Basic properties 121 4.3 Reduced decompositions 129 4.4 The normalization map 137 4.5 Chevalley’s multiplicity formula 140 4.6 Deodhar’s Lemma 145 Appendix A Cohen-Macaulay Properties 149 Appendix B Normality of Schubert varieties 166 Appendix C Standard Monomial Theory 173 1 Introduction 173 2 First Basis Theorem 174 3 Main Theorems for Classical Type 179 4 The Exceptional Groups and the Kac-Moody Groups 185 5 Applications 201 References 217 Bibliography 220 Notation 223 Index 224 Symbols 226 Texts and Readings in Mathematics 228 Provides an introduction to what has come to be known as Standard Monomial Theory (SMT). SMT deals with the construction of nice bases of finite dimensional irreducible representations of semi-simple algebraic groups or, in geometric terms, nice bases of coordinate rings of flag varieties (and their Schubert subvarieties) associated to these groups. Front Matter....Pages i-xvi Schubert Varieties in the Grassmannian....Pages 1-53 Standard monomial theory on SLn(k)/Q....Pages 55-80 Applications....Pages 81-105 Schubert varieties in G/Q....Pages 107-137 Back Matter....Pages 139-224