Flag Varieties

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Flag Varieties

V Lakshmibai,Justin Brown
Flag Varieties Book PDF

Author : V Lakshmibai,Justin Brown
Publisher : Springer
Page : 312 pages
File Size : 43,7 Mb
Release : 2018-06-26
Category : Mathematics
ISBN : 9789811313936

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Flag Varieties by V Lakshmibai,Justin Brown Pdf

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

Shrawan Kumar
Kac Moody Groups their Flag Varieties and Representation Theory Book PDF

Author : Shrawan Kumar
Publisher : Springer Science & Business Media
Page : 613 pages
File Size : 52,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461201052

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Kac-Moody Groups, their Flag Varieties and Representation Theory by Shrawan Kumar Pdf

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.

Affine Flag Varieties and Quantum Symmetric Pairs

Zhaobing Fan,Chun-Ju Lai,Yiqiang Li,Li Luo,Weiqiang Wang
Affine Flag Varieties and Quantum Symmetric Pairs Book PDF

Author : Zhaobing Fan,Chun-Ju Lai,Yiqiang Li,Li Luo,Weiqiang Wang
Publisher : American Mathematical Soc.
Page : 123 pages
File Size : 55,5 Mb
Release : 2020-09-28
Category : Mathematics
ISBN : 9781470441753

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Affine Flag Varieties and Quantum Symmetric Pairs by Zhaobing Fan,Chun-Ju Lai,Yiqiang Li,Li Luo,Weiqiang Wang Pdf

The quantum groups of finite and affine type $A$ admit geometric realizations in terms of partial flag varieties of finite and affine type $A$. Recently, the quantum group associated to partial flag varieties of finite type $B/C$ is shown to be a coideal subalgebra of the quantum group of finite type $A$.

Topics in Cohomological Studies of Algebraic Varieties

Piotr Pragacz
Topics in Cohomological Studies of Algebraic Varieties Book PDF

Author : Piotr Pragacz
Publisher : Springer Science & Business Media
Page : 321 pages
File Size : 43,8 Mb
Release : 2006-03-30
Category : Mathematics
ISBN : 9783764373429

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Topics in Cohomological Studies of Algebraic Varieties by Piotr Pragacz Pdf

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 Geometry of the Flag Variety

William M. McGovern
Representation Theory and Geometry of the Flag Variety Book PDF

Author : William M. McGovern
Publisher : Walter de Gruyter GmbH & Co KG
Page : 136 pages
File Size : 48,9 Mb
Release : 2022-11-07
Category : Mathematics
ISBN : 9783110766943

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Representation Theory and Geometry of the Flag Variety by William M. McGovern Pdf

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.

Cohomology of Vector Bundles and Syzygies

Jerzy Weyman
Cohomology of Vector Bundles and Syzygies Book PDF

Author : Jerzy Weyman
Publisher : Cambridge University Press
Page : 404 pages
File Size : 49,6 Mb
Release : 2003-06-09
Category : Mathematics
ISBN : 0521621976

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Cohomology of Vector Bundles and Syzygies by Jerzy Weyman Pdf

The central theme of this book is an exposition of the geometric technique of calculating syzygies. It is written from a point of view of commutative algebra, and without assuming any knowledge of representation theory the calculation of syzygies of determinantal varieties is explained. The starting point is a definition of Schur functors, and these are discussed from both an algebraic and geometric point of view. Then a chapter on various versions of Bott's Theorem leads on to a careful explanation of the technique itself, based on a description of the direct image of a Koszul complex. Applications to determinantal varieties follow, plus there are also chapters on applications of the technique to rank varieties for symmetric and skew symmetric tensors of arbitrary degree, closures of conjugacy classes of nilpotent matrices, discriminants and resultants. Numerous exercises are included to give the reader insight into how to apply this important method.

Schubert Varieties and Degeneracy Loci

William Fulton,Piotr Pragacz
Schubert Varieties and Degeneracy Loci Book PDF

Author : William Fulton,Piotr Pragacz
Publisher : Springer
Page : 158 pages
File Size : 48,9 Mb
Release : 2006-11-13
Category : Mathematics
ISBN : 9783540698043

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Schubert Varieties and Degeneracy Loci by William Fulton,Piotr Pragacz Pdf

Schubert varieties and degeneracy loci have a long history in mathematics, starting from questions about loci of matrices with given ranks. These notes, from a summer school in Thurnau, aim to give an introduction to these topics, and to describe recent progress on these problems. There are interesting interactions with the algebra of symmetric functions and combinatorics, as well as the geometry of flag manifolds and intersection theory and algebraic geometry.

Cohomology of Vector Bundles and Syzygies

Jerzy Weyman
Cohomology of Vector Bundles and Syzygies Book PDF

Author : Jerzy Weyman
Publisher : Cambridge University Press
Page : 404 pages
File Size : 49,8 Mb
Release : 2003-06-09
Category : Mathematics
ISBN : 0521621976

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Cohomology of Vector Bundles and Syzygies by Jerzy Weyman Pdf

The central theme of this book is an exposition of the geometric technique of calculating syzygies. It is written from a point of view of commutative algebra, and without assuming any knowledge of representation theory the calculation of syzygies of determinantal varieties is explained. The starting point is a definition of Schur functors, and these are discussed from both an algebraic and geometric point of view. Then a chapter on various versions of Bott's Theorem leads on to a careful explanation of the technique itself, based on a description of the direct image of a Koszul complex. Applications to determinantal varieties follow, plus there are also chapters on applications of the technique to rank varieties for symmetric and skew symmetric tensors of arbitrary degree, closures of conjugacy classes of nilpotent matrices, discriminants and resultants. Numerous exercises are included to give the reader insight into how to apply this important method.

Frobenius Splitting Methods in Geometry and Representation Theory

Michel Brion,Shrawan Kumar
Frobenius Splitting Methods in Geometry and Representation Theory Book PDF

Author : Michel Brion,Shrawan Kumar
Publisher : Springer Science & Business Media
Page : 259 pages
File Size : 42,5 Mb
Release : 2007-08-08
Category : Mathematics
ISBN : 9780817644055

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Frobenius Splitting Methods in Geometry and Representation Theory by Michel Brion,Shrawan Kumar Pdf

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.

Singular Loci of Schubert Varieties

Sara Sarason,V. Lakshmibai
Singular Loci of Schubert Varieties Book PDF

Author : Sara Sarason,V. Lakshmibai
Publisher : Springer Science & Business Media
Page : 254 pages
File Size : 45,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461213246

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Singular Loci of Schubert Varieties by Sara Sarason,V. Lakshmibai Pdf

"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.

Standard Monomial Theory

V. Lakshmibai,K. N. Raghavan
Standard Monomial Theory Book PDF

Author : V. Lakshmibai,K. N. Raghavan
Publisher : Springer Science & Business Media
Page : 271 pages
File Size : 48,8 Mb
Release : 2007-12-23
Category : Mathematics
ISBN : 9783540767572

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Standard Monomial Theory by V. Lakshmibai,K. N. Raghavan Pdf

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.

The Grassmannian Variety

V. Lakshmibai,Justin Brown
The Grassmannian Variety Book PDF

Author : V. Lakshmibai,Justin Brown
Publisher : Springer
Page : 172 pages
File Size : 46,6 Mb
Release : 2015-09-25
Category : Mathematics
ISBN : 9781493930821

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The Grassmannian Variety by V. Lakshmibai,Justin Brown Pdf

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.

Symmetric Functions, Schubert Polynomials and Degeneracy Loci

Laurent Manivel
Symmetric Functions Schubert Polynomials and Degeneracy Loci Book PDF

Author : Laurent Manivel
Publisher : American Mathematical Soc.
Page : 180 pages
File Size : 54,7 Mb
Release : 2001
Category : Computers
ISBN : 0821821547

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Symmetric Functions, Schubert Polynomials and Degeneracy Loci by Laurent Manivel Pdf

This text grew out of an advanced course taught by the author at the Fourier Institute (Grenoble, France). It serves as an introduction to the combinatorics of symmetric functions, more precisely to Schur and Schubert polynomials. Also studied is the geometry of Grassmannians, flag varieties, and especially, their Schubert varieties. This book examines profound connections that unite these two subjects. The book is divided into three chapters. The first is devoted to symmetricfunctions and especially to Schur polynomials. These are polynomials with positive integer coefficients in which each of the monomials correspond to a Young tableau with the property of being ``semistandard''. The second chapter is devoted to Schubert polynomials, which were discovered by A. Lascoux andM.-P. Schutzenberger who deeply probed their combinatorial properties. It is shown, for example, that these polynomials support the subtle connections between problems of enumeration of reduced decompositions of permutations and the Littlewood-Richardson rule, a particularly efficacious version of which may be derived from these connections. The final chapter is geometric. It is devoted to Schubert varieties, subvarieties of Grassmannians, and flag varieties defined by certain incidenceconditions with fixed subspaces. This volume makes accessible a number of results, creating a solid stepping stone for scaling more ambitious heights in the area. The author's intent was to remain elementary: The first two chapters require no prior knowledge, the third chapter uses some rudimentary notionsof topology and algebraic geometry. For this reason, a comprehensive appendix on the topology of algebraic varieties is provided. This book is the English translation of a text previously published in French.

Ulrich Bundles

Laura Costa,Rosa María Miró-Roig,Joan Pons-Llopis
Ulrich Bundles Book PDF

Author : Laura Costa,Rosa María Miró-Roig,Joan Pons-Llopis
Publisher : Walter de Gruyter GmbH & Co KG
Page : 282 pages
File Size : 42,9 Mb
Release : 2021-05-10
Category : Mathematics
ISBN : 9783110647686

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Ulrich Bundles by Laura Costa,Rosa María Miró-Roig,Joan Pons-Llopis Pdf

The goal of this book is to cover the active developments of arithmetically Cohen-Macaulay and Ulrich bundles and related topics in the last 30 years, and to present relevant techniques and multiple applications of the theory of Ulrich bundles to a wide range of problems in algebraic geometry as well as in commutative algebra.

Grassmann and Stiefel Varieties over Composition Algebras

Marek Golasiński,Francisco Gómez Ruiz
Grassmann and Stiefel Varieties over Composition Algebras Book PDF

Author : Marek Golasiński,Francisco Gómez Ruiz
Publisher : Springer Nature
Page : 342 pages
File Size : 44,9 Mb
Release : 2023-09-17
Category : Mathematics
ISBN : 9783031364051

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Grassmann and Stiefel Varieties over Composition Algebras by Marek Golasiński,Francisco Gómez Ruiz Pdf

This monograph deals with matrix manifolds, i.e., manifolds for which there is a natural representation of their elements as matrix arrays. Classical matrix manifolds (Stiefel, Grassmann and flag manifolds) are studied in a more general setting. It provides tools to investigate matrix varieties over Pythagorean formally real fields. The presentation of the book is reasonably self-contained. It contains a number of nontrivial results on matrix manifolds useful for people working not only in differential geometry and Riemannian geometry but in other areas of mathematics as well. It is also designed to be readable by a graduate student who has taken introductory courses in algebraic and differential geometry.