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Representation Theory And Geometry Of The Flag Variety
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Book Synopsis Representation Theory and Geometry of the Flag Variety by : William M. McGovern
Download or read book Representation Theory and Geometry of the Flag Variety written by William M. McGovern and published by Walter de Gruyter GmbH & Co KG. This book was released on 2022-11-07 with total page 136 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Flag Varieties written by V Lakshmibai and published by Springer. This book was released on 2018-06-26 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Flag Varieties by : Justin Brown V Lakshmibai
Download or read book Flag Varieties written by Justin Brown V Lakshmibai and published by . This book was released on 2018 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Kac-Moody Groups, their Flag Varieties and Representation Theory by : Shrawan Kumar
Download or read book Kac-Moody Groups, their Flag Varieties and Representation Theory written by Shrawan Kumar and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 613 pages. Available in PDF, EPUB and Kindle. Book excerpt: Kac-Moody Lie algebras 9 were introduced in the mid-1960s independently by V. Kac and R. Moody, generalizing the finite-dimensional semisimple Lie alge bras which we refer to as the finite case. The theory has undergone tremendous developments in various directions and connections with diverse areas abound, including mathematical physics, so much so that this theory has become a stan dard tool in mathematics. A detailed treatment of the Lie algebra aspect of the theory can be found in V. Kac's book [Kac-90l This self-contained work treats the algebro-geometric and the topological aspects of Kac-Moody theory from scratch. The emphasis is on the study of the Kac-Moody groups 9 and their flag varieties XY, including their detailed construction, and their applications to the representation theory of g. In the finite case, 9 is nothing but a semisimple Y simply-connected algebraic group and X is the flag variety 9 /Py for a parabolic subgroup p y C g.
Book Synopsis Frobenius Splitting Methods in Geometry and Representation Theory by : Michel Brion
Download or read book Frobenius Splitting Methods in Geometry and Representation Theory written by Michel Brion and published by Springer Science & Business Media. This book was released on 2007-08-08 with total page 259 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Topics in Cohomological Studies of Algebraic Varieties by : Piotr Pragacz
Download or read book Topics in Cohomological Studies of Algebraic Varieties written by Piotr Pragacz and published by Springer Science & Business Media. This book was released on 2006-03-30 with total page 321 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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
Book Synopsis Representation Theory and Complex Geometry by : Neil Chriss
Download or read book Representation Theory and Complex Geometry written by Neil Chriss and published by . This book was released on 1997 with total page 520 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is an attempt to provide an overview of some of the recent advances in representation theory from a geometric standpoint. A geometrically-oriented treatment is very timely and has long been desired, especially since the discovery of D-modules in the early '80s and the quiver approach to quantum groups in the early '90s.
Book Synopsis Singular Loci of Schubert Varieties by : Sara Sarason
Download or read book Singular Loci of Schubert Varieties written by Sara Sarason and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 254 pages. Available in PDF, EPUB and Kindle. Book excerpt: "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.
Book Synopsis Lie Theory and Geometry by : Jean-Luc Brylinski
Download or read book Lie Theory and Geometry written by Jean-Luc Brylinski and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 629 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume, dedicated to Bertram Kostant on the occasion of his 65th birthday, is a collection of 22 invited papers by leading mathematicians working in Lie theory, geometry, algebra, and mathematical physics. Kostant’s fundamental work in all these areas has provided deep new insights and connections, and has created new fields of research. The papers gathered here present original research articles as well as expository papers, broadly reflecting the range of Kostant’s work.
Book Synopsis Representations of Algebraic Groups by : Jens Carsten Jantzen
Download or read book Representations of Algebraic Groups written by Jens Carsten Jantzen and published by American Mathematical Soc.. This book was released on 2003 with total page 594 pages. Available in PDF, EPUB and Kindle. Book excerpt: Gives an introduction to the general theory of representations of algebraic group schemes. This title deals with representation theory of reductive algebraic groups and includes topics such as the description of simple modules, vanishing theorems, Borel-Bott-Weil theorem and Weyl's character formula, and Schubert schemes and lne bundles on them.
Book Synopsis Geometry of Moduli Spaces and Representation Theory by : Roman Bezrukavnikov
Download or read book Geometry of Moduli Spaces and Representation Theory written by Roman Bezrukavnikov and published by American Mathematical Soc.. This book was released on 2017-12-15 with total page 436 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is based on lectures given at the Graduate Summer School of the 2015 Park City Mathematics Institute program “Geometry of moduli spaces and representation theory”, and is devoted to several interrelated topics in algebraic geometry, topology of algebraic varieties, and representation theory. Geometric representation theory is a young but fast developing research area at the intersection of these subjects. An early profound achievement was the famous conjecture by Kazhdan–Lusztig about characters of highest weight modules over a complex semi-simple Lie algebra, and its subsequent proof by Beilinson-Bernstein and Brylinski-Kashiwara. Two remarkable features of this proof have inspired much of subsequent development: intricate algebraic data turned out to be encoded in topological invariants of singular geometric spaces, while proving this fact required deep general theorems from algebraic geometry. Another focus of the program was enumerative algebraic geometry. Recent progress showed the role of Lie theoretic structures in problems such as calculation of quantum cohomology, K-theory, etc. Although the motivation and technical background of these constructions is quite different from that of geometric Langlands duality, both theories deal with topological invariants of moduli spaces of maps from a target of complex dimension one. Thus they are at least heuristically related, while several recent works indicate possible strong technical connections. The main goal of this collection of notes is to provide young researchers and experts alike with an introduction to these areas of active research and promote interaction between the two related directions.
Download or read book Young Tableaux written by William Fulton and published by Cambridge University Press. This book was released on 1997 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt: Describes combinatorics involving Young tableaux and their uses in representation theory and algebraic geometry.
Book Synopsis The Grassmannian Variety by : V. Lakshmibai
Download or read book The Grassmannian Variety written by V. Lakshmibai and published by Springer. This book was released on 2015-09-25 with total page 172 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis The Penrose Transform by : Robert J. Baston
Download or read book The Penrose Transform written by Robert J. Baston and published by Courier Dover Publications. This book was released on 2016-10-28 with total page 256 pages. Available in PDF, EPUB and Kindle. Book excerpt: Geared toward students of physics and mathematics; presupposes no familiarity with twistor theory. "A huge amount of information, well organized and condensed into less than 200 pages." — Mathematical Reviews. 1989 edition.
Book Synopsis Standard Monomial Theory by : V. Lakshmibai
Download or read book Standard Monomial Theory written by V. Lakshmibai and published by Springer Science & Business Media. This book was released on 2007-12-23 with total page 271 pages. Available in PDF, EPUB and Kindle. Book excerpt: Schubert varieties provide an inductive tool for studying flag varieties. This book is mainly a detailed account of a particularly interesting instance of their occurrence: namely, in relation to classical invariant theory. More precisely, it is about the connection between the first and second fundamental theorems of classical invariant theory on the one hand and standard monomial theory for Schubert varieties in certain special flag varieties on the other.
Book Synopsis Geometry and Representation Theory of Real and p-adic groups by : Juan Tirao
Download or read book Geometry and Representation Theory of Real and p-adic groups written by Juan Tirao and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 330 pages. Available in PDF, EPUB and Kindle. Book excerpt: The representation theory of Lie groups plays a central role in both clas sical and recent developments in many parts of mathematics and physics. In August, 1995, the Fifth Workshop on Representation Theory of Lie Groups and its Applications took place at the Universidad Nacional de Cordoba in Argentina. Organized by Joseph Wolf, Nolan Wallach, Roberto Miatello, Juan Tirao, and Jorge Vargas, the workshop offered expository courses on current research, and individual lectures on more specialized topics. The present vol ume reflects the dual character of the workshop. Many of the articles will be accessible to graduate students and others entering the field. Here is a rough outline of the mathematical content. (The editors beg the indulgence of the readers for any lapses in this preface in the high standards of historical and mathematical accuracy that were imposed on the authors of the articles. ) Connections between flag varieties and representation theory for real re ductive groups have been studied for almost fifty years, from the work of Gelfand and Naimark on principal series representations to that of Beilinson and Bernstein on localization. The article of Wolf provides a detailed introduc tion to the analytic side of these developments. He describes the construction of standard tempered representations in terms of square-integrable partially harmonic forms (on certain real group orbits on a flag variety), and outlines the ingredients in the Plancherel formula. Finally, he describes recent work on the complex geometry of real group orbits on partial flag varieties.
Book Synopsis Affine Flag Manifolds and Principal Bundles by : Alexander Schmitt
Download or read book Affine Flag Manifolds and Principal Bundles written by Alexander Schmitt and published by Springer Science & Business Media. This book was released on 2011-01-28 with total page 298 pages. Available in PDF, EPUB and Kindle. Book excerpt: Affine flag manifolds are infinite dimensional versions of familiar objects such as Graßmann varieties. The book features lecture notes, survey articles, and research notes - based on workshops held in Berlin, Essen, and Madrid - explaining the significance of these and related objects (such as double affine Hecke algebras and affine Springer fibers) in representation theory (e.g., the theory of symmetric polynomials), arithmetic geometry (e.g., the fundamental lemma in the Langlands program), and algebraic geometry (e.g., affine flag manifolds as parameter spaces for principal bundles). Novel aspects of the theory of principal bundles on algebraic varieties are also studied in the book.
