This book gives a comprehensive treatment of the Grassmannian varieties
and their Schubert subvarieties, focusing on the geometric and
representation-theoretic aspects of Grassmannian varieties. Research of
Grassmannian varieties is centered at the crossroads of commutative
algebra, algebraic geometry, representation theory, and combinatorics.
Therefore, this text uniquely presents an exciting playing field for
graduate students and researchers in mathematics, physics, and computer
science, to expand their knowledge in the field of algebraic geometry.
The standard monomial theory (SMT) for the Grassmannian varieties and
their Schubert subvarieties are introduced and the text presents some
important applications of SMT including the Cohen-Macaulay property,
normality, unique factoriality, Gorenstein property, singular loci of
Schubert varieties, toric degenerations of Schubert varieties, and the
relationship between Schubert varieties and classical invariant theory.
This text would serve well as a reference book for a graduate work on
Grassmannian varieties and would be an excellent supplementary text for
several courses including those in geometry of spherical varieties,
Schubert varieties, advanced topics in geometric and differential
topology, representation theory of compact and reductive groups, Lie
theory, toric varieties, geometric representation theory, and
singularity theory. The reader should have some familiarity with
commutative algebra and algebraic geometry.
