# Flag Varieties

## An Interplay of Geometry, Combinatorics, and Representation Theory

Part of the Texts and Readings in Mathematics book series (TRIM, volume 53)

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Part of the Texts and Readings in Mathematics book series (TRIM, volume 53)

This book discusses the importance of flag varieties in geometric objects and elucidates its richness as interplay of geometry, combinatorics and 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 book also discusses the representation theory of symmetric groups. In the area of algebraic geometry, the book gives a detailed account of the Grassmannian varieties, flag varieties, and their Schubert subvarieties. Many of the geometric results admit elegant combinatorial description because of the root system connections, 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. Consequently, this book includes standard monomial theory and some important applications—singular loci of Schubert varieties, toric degenerations of Schubert varieties, and the relationship between Schubert varieties and classical invariant theory. The two recent results on Schubert varieties in the Grassmannian have also been included in this book. 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.

Semisimple Rings Finite Groups Symmetric Group Symmetric Polynomials Schur-Weyl Duality Semisimple Lie Algebras Algebraic Groups Reductive Groups

- DOI https://doi.org/10.1007/978-981-13-1393-6
- Copyright Information Springer Nature Singapore Pte Ltd. 2018 and Hindustan Book Agency 2018 2018
- Publisher Name Springer, Singapore
- eBook Packages Mathematics and Statistics
- Online ISBN 978-981-13-1393-6
- Series Print ISSN 2366-8717
- Series Online ISSN 2366-8725
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