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.
