Series:

Progress in Mathematics

1 primary work • 2 total works

Book 182

Cover of Singular Loci of Schubert Varieties

"Singular Loci of Schubert Varieties" is a unique work at the crossroads of representation theory, algebraic geometry, and combinatorics. Over the past 20 years, many research articles have been written on the subject in notable journals. In this work, Billey and Lakshmibai have recreated and restructured the various theories and approaches of those articles and present a clearer understanding of this important subdiscipline of Schubert varieties - namely singular loci. The main focus, therefore, is on the computations for the singular loci of Schubert varieties and corresponding tangent spaces. The methods used include standard monomial theory, the nil Hecke ring, and Kazhdan-Lusztig theory. New results are presented with sufficient examples to emphasize key points. A comprehensive bibliography, index, and tables - the latter not to be found elsewhere in the mathematics literature - round out this concise work. After a good introduction giving background material, the topics are presented in a systematic fashion to engage a wide readership of researchers and graduate students.

v. 257

Cover of Schubert Varieties
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.