# Flag Varieties: An Interplay of Geometry, Combinatorics, and Representation Theory: Second Edition

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*V. Lakshmibai; Justin Brown*

A publication of Hindustan Book Agency

Flag varieties are important geometric objects. Because of
their richness in geometry, combinatorics, and representation theory,
flag varieties may be described as an interplay of all three of these
fields.

This book gives a detailed account of this interplay. In the area
of representation theory, the book presents a discussion on the
representation theory of complex semisimple Lie algebras as well as
the representation theory of semisimple algebraic groups; in addition,
the representation theory of symmetric groups is also discussed. In
the area of algebraic geometry, the book gives a detailed account of
the Grassmannian varieties, flag varieties, and their Schubert
subvarieties. Because of the root system connections, many of the
geometric results admit elegant combinatorial description, a typical
example being the description of the singular locus of a Schubert
variety. This discussion is carried out as a consequence of standard
monomial theory (abbreviated SMT). Thus, the book includes SMT and
some important applications—singular loci of Schubert varieties,
toric degenerations of Schubert varieties, and the relationship
between Schubert varieties and classical invariant theory.

In the second edition, two recent results on Schubert varieties in the
Grassmannian have been added. The first result gives a free resolution
of certain Schubert singularities. The second result is about certain
Levi subgroup actions on Schubert varieties in the Grassmannian and
derives some interesting geometric and representation-theoretic
consequences.

A publication of Hindustan Book Agency; distributed within the Americas by the American Mathematical Society. Maximum discount of 20% for all commercial channels.