Hindustan Book Agency 2009; 288 pp; hardcover ISBN10: 8185931925 ISBN13: 9788185931920 List Price: US$52 Member Price: US$41.60 Order Code: HIN/40
 Flag varieties are important geometric objects and their study involves an interplay of geometry, combinatorics, and representation theory. This book is a detailed account of this interplay. In the area of representation theory, the book discusses complex semisimple Lie algebras and semisimple algebraic groups; in addition, the representation theory of symmetric groups is discussed. In the area of algebraic geometry, the book explains in detail Grassmannian varieties, flag varieties, and their Schubert subvarieties. Because of the connections with root systems, many of the geometric results admit elegant combinatorial description, a typical example being the description of the singular locus of a Schubert variety. This is shown to be a consequence of standard monomial theory (abbreviated SMT). Thus the book includes SMT and some important applicationssingular loci of Schubert varieties, toric degenerations of Schubert varieties, and the relationship between Schubert varieties and classical invariant theory. A publication of Hindustan Book Agency; distributed within the Americas by the American Mathematical Society. Maximum discount of 20% for all commercial channels. Readership Graduate students and research mathematicians interested in flag varieties. Table of Contents  Introduction
 Preliminaries
 Structure theory of semisimple rings
 Representation theory of finite groups
 Representation theory of the symmetric group
 Symmetric polynomials
 SchurWeyl duality and the relationship between representations of \(Sd\) and \(GL_n (C)\)
 Structure theory of complex semisimple Lie algebras
 Representation theory of complex semisimple Lie algebras
 Generalities on algebraic groups
 Structure theory of reductive groups
 Representation theory of semisimple algebraic groups
 Geometry of the grassmannian, flag and their Schubert varieties via standard monomial theory
 Singular locus of a Schubert variety in the flag variety \(SL_n/B\)
 Applications
 Appendix: Chevalley groups
 Bibliography
 List of symbols
 Index
