Over the past two decades, and more intensely in recent years, the algebro-geometric study of Schubert Varieties has had considerable impact on the theory of algebraic groups. One of the most interesting developments in the theory has been the construction of natural bases of representations of the full linear group $GL(n)$, the orthogonal group, and the symplectic group. This construction gives character formulas of these representations which are quite different in spirit from the famous character formulas of H. Weyl. In fact, they connect to monomial theory and the work of Hodge which was done more than fifty years ago, and to the very recent developments in path models, Frobenius splittings, and quantum groups. Written by three of the world''s leading mathematicians in algebraic geometry, group theory, and combinatorics, this excellent self- contained exposition on Schubert Varieties unfolds systematically, from relevant introductory material on commutative algebra and algebraic geometry. First-rate text for a graduate course or for self-study.

Preface.- Preliminaries.- SMT for Graßmann varieties.- Combinatorics of Weyl groups and Schubert varieties.- SMT for flag varieties.- Path models of representations.- Frobenius map and Path Basis.- Path Models.- Quantum Demazure Modules.- Standard Monomial Basis.- Applications Standard Monomial Basis.- References.- Index.