## Texts and Readings in Mathematics

Flag Varieties - An Interplay of Geometry, Combinatorics and Representation Theory

V. Lakshmibai and Justin Brown

Flag varieties are important geometric objects and their study involves an interplay of geometry, combinatorics, and representation theory. This book is detailed account of this interplay.

In the area of representation theory, the book presents a discussion of complex semisimple Lie algebras and 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 connections with root systems, many of the geometric results admit elegant combinatorial description, a typical example being the description of the singular locus of a Schubert variety. This is shown to be 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.

Contents

1. Preliminaries 2. Structure Theory of Semisimple Rings

3. Representation Theory of Finite Groups

4. Representation Theory of the Symmetric Group

5. Symmetric Polynomials

6. Schur-Weyl Duality and the Relationship Between Representations of Sd and GLn (C)

7. Structure Theory of Complex Semisimple Lie Algebras

8. Representation Theory of Complex Semisimple Lie Algebras

9. Generalities on Algebraic Groups

10. Structure Theory of Reductive Groups

11. Representation Theory of Semisimple Algebraic Groups

12. Geometry of the Grassmannian, Flag and their Schubert Varieties via Standard Monomial Theory

13. Singular Locus of a Schubert Variety in the Flag Variety SLn/B

14. Applications.

Appendix: Chevalley groups

Bibliography

List of symbols

Index

Texts and Readings in Mathematics/ 53

2009, 9788185931920, 288 pages, hard cover, Rs. 500.00

2009, 9788185931920, 288 pages, hard cover, Rs. 500.00