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. TOC:Preface * Preliminaries * Algebraic Groups * Generalities on G/B and
G/Q * The Grassmannian * The Flag Variety * Desingularization of
Schubert Varieties * Path Models * Quantum Demazure Modules * Standard
Monomial Basis * Applications Standard Monomial Basis * References *
Index
