Flag Varieties
Flag Varieties

Author: V Lakshmibai

Publisher: Springer

Published: 2018-06-26

Total Pages: 315

ISBN-13: 9811313938

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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.

Kac-Moody Groups, their Flag Varieties and Representation Theory
Kac-Moody Groups, their Flag Varieties and Representation Theory

Author: Shrawan Kumar

Publisher: Springer Science & Business Media

Published: 2002-09-10

Total Pages: 630

ISBN-13: 9780817642273

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"Most of these topics appear here for the first time in book form. Many of them are interesting even in the classical case of semi-simple algebraic groups. Some appendices recall useful results from other areas, so the work may be considered self-contained, although some familiarity with semi-simple Lie algebras or algebraic groups is helpful. It is clear that this book is a valuable reference for all those interested in flag varieties and representation theory in the semi-simple or Kac-Moody case." —MATHEMATICAL REVIEWS "A lot of different topics are treated in this monumental work. . . . many of the topics of the book will be useful for those only interested in the finite-dimensional case. The book is self contained, but is on the level of advanced graduate students. . . . For the motivated reader who is willing to spend considerable time on the material, the book can be a gold mine. " —ZENTRALBLATT MATH

Representation Theory and Geometry of the Flag Variety
Representation Theory and Geometry of the Flag Variety

Author: William M. McGovern

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2022-11-07

Total Pages: 136

ISBN-13: 3110766949

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This comprehensive reference begins with a review of the basics followed by a presentation of flag varieties and finite- and infinite-dimensional representations in classical types and subvarieties of flag varieties and their singularities. Associated varieties and characteristic cycles are covered as well and Kazhdan-Lusztig polynomials are treated. The coverage concludes with a discussion of pattern avoidance and singularities and some recent results on Springer fibers.

Frobenius Splitting Methods in Geometry and Representation Theory
Frobenius Splitting Methods in Geometry and Representation Theory

Author: Michel Brion

Publisher: Springer Science & Business Media

Published: 2007-08-08

Total Pages: 259

ISBN-13: 0817644059

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Systematically develops the theory of Frobenius splittings and covers all its major developments. Concise, efficient exposition unfolds from basic introductory material on Frobenius splittings—definitions, properties and examples—to cutting edge research.

Young Tableaux
Young Tableaux

Author: William Fulton

Publisher: Cambridge University Press

Published: 1997

Total Pages: 276

ISBN-13: 9780521567244

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Describes combinatorics involving Young tableaux and their uses in representation theory and algebraic geometry.

Topics in Cohomological Studies of Algebraic Varieties
Topics in Cohomological Studies of Algebraic Varieties

Author: Piotr Pragacz

Publisher: Springer Science & Business Media

Published: 2006-03-30

Total Pages: 321

ISBN-13: 3764373423

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The articles in this volume study various cohomological aspects of algebraic varieties: - characteristic classes of singular varieties; - geometry of flag varieties; - cohomological computations for homogeneous spaces; - K-theory of algebraic varieties; - quantum cohomology and Gromov-Witten theory. The main purpose is to give comprehensive introductions to the above topics through a series of "friendly" texts starting from a very elementary level and ending with the discussion of current research. In the articles, the reader will find classical results and methods as well as new ones. Numerous examples will help to understand the mysteries of the cohomological theories presented. The book will be a useful guide to research in the above-mentioned areas. It is adressed to researchers and graduate students in algebraic geometry, algebraic topology, and singularity theory, as well as to mathematicians interested in homogeneous varieties and symmetric functions. Most of the material exposed in the volume has not appeared in books before. Contributors: Paolo Aluffi Michel Brion Anders Skovsted Buch Haibao Duan Ali Ulas Ozgur Kisisel Piotr Pragacz Jörg Schürmann Marek Szyjewski Harry Tamvakis

Representation Theory and Complex Geometry
Representation Theory and Complex Geometry

Author: Neil Chriss

Publisher: Birkhauser

Published: 1997

Total Pages: 495

ISBN-13: 0817637923

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This volume provides an overview of modern advances in representation theory from a geometric standpoint. The techniques developed are quite general and can be applied to other areas such as quantum groups, affine Lie groups, and quantum field theory.

Singular Loci of Schubert Varieties
Singular Loci of Schubert Varieties

Author: Sara Sarason

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 254

ISBN-13: 146121324X

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"Singular Loci of Schubert Varieties" is a unique work at the crossroads of representation theory, algebraic geometry, and combinatorics. Over the past 20 years, many research articles have been written on the subject in notable journals. In this work, Billey and Lakshmibai have recreated and restructured the various theories and approaches of those articles and present a clearer understanding of this important subdiscipline of Schubert varieties – namely singular loci. The main focus, therefore, is on the computations for the singular loci of Schubert varieties and corresponding tangent spaces. The methods used include standard monomial theory, the nil Hecke ring, and Kazhdan-Lusztig theory. New results are presented with sufficient examples to emphasize key points. A comprehensive bibliography, index, and tables – the latter not to be found elsewhere in the mathematics literature – round out this concise work. After a good introduction giving background material, the topics are presented in a systematic fashion to engage a wide readership of researchers and graduate students.

Representation Theory and Geometry of the Flag Variety
Representation Theory and Geometry of the Flag Variety

Author: William M McGovern

Publisher:

Published: 2022-12-28

Total Pages: 0

ISBN-13: 9783110766905

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The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 30 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics. While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. Please submit any book proposals to Niels Jacob.

The Grassmannian Variety
The Grassmannian Variety

Author: V. Lakshmibai

Publisher: Springer

Published: 2015-09-25

Total Pages: 174

ISBN-13: 1493930826

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This book gives a comprehensive treatment of the Grassmannian varieties and their Schubert subvarieties, focusing on the geometric and representation-theoretic aspects of Grassmannian varieties. Research of Grassmannian varieties is centered at the crossroads of commutative algebra, algebraic geometry, representation theory, and combinatorics. Therefore, this text uniquely presents an exciting playing field for graduate students and researchers in mathematics, physics, and computer science, to expand their knowledge in the field of algebraic geometry. The standard monomial theory (SMT) for the Grassmannian varieties and their Schubert subvarieties are introduced and the text presents some important applications of SMT including the Cohen–Macaulay property, normality, unique factoriality, Gorenstein property, singular loci of Schubert varieties, toric degenerations of Schubert varieties, and the relationship between Schubert varieties and classical invariant theory. This text would serve well as a reference book for a graduate work on Grassmannian varieties and would be an excellent supplementary text for several courses including those in geometry of spherical varieties, Schubert varieties, advanced topics in geometric and differential topology, representation theory of compact and reductive groups, Lie theory, toric varieties, geometric representation theory, and singularity theory. The reader should have some familiarity with commutative algebra and algebraic geometry.