Flag varieties

Flag varieties constitute an important class of homogeneous spaces. Because of
their rich geometry and combinatorics, they represent fundamental objects in
the areas of Algebraic Geometry, Algebraic Groups and Representation Theory.
This book provides an introduction to the subject, and presents the interplay
of flag varieties among Geometry, Combinatorics and Representation Theory. The
Schubert subvarieties provide a powerful inductive machinery for the study of
flag varieties. The central theme of this book is the theory of Schubert
varieties - their geometric properties, ideal theory, singularity theory. This
book also presents the relationship between Schubert varieties and certain
affine varieties - classical determinantal varieties, ladder determinantal
varieties, quiver varieties. varieties of complexes, certain affine toric
varieties. This book includes a detailed treatment of the classical
determinantal varieties, their normality, Cohen-Macaulayness, singular loci,
relationship to classical invariant theory : the treatment uses their
relationship to Schubert varieties.