Standard Monomial Theory

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Introduction to the Theory of Standard Monomials

Author : C. S. Seshadri
Publisher : Springer
ISBN 13 : 9811018138
Total Pages : 229 pages
Book Rating : 4.8/5 (11 download)

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Book Synopsis Introduction to the Theory of Standard Monomials by : C. S. Seshadri

Download or read book Introduction to the Theory of Standard Monomials written by C. S. Seshadri and published by Springer. This book was released on 2016-08-22 with total page 229 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book is a reproduction of a course of lectures delivered by the author in 1983-84 which appeared in the Brandeis Lecture Notes series. The aim of this course was to give an introduction to the series of papers by concentrating on the case of the full linear group. In recent years, there has been great progress in standard monomial theory due to the work of Peter Littelmann. The author’s lectures (reproduced in this book) remain an excellent introduction to standard monomial theory. Standard monomial theory deals with the construction of nice bases of finite dimensional irreducible representations of semi-simple algebraic groups or, in geometric terms, nice bases of coordinate rings of flag varieties (and their Schubert subvarieties) associated with these groups. Besides its intrinsic interest, standard monomial theory has applications to the study of the geometry of Schubert varieties. Standard monomial theory has its origin in the work of Hodge, giving bases of the coordinate rings of the Grassmannian and its Schubert subvarieties by “standard monomials”. In its modern form, standard monomial theory was developed by the author in a series of papers written in collaboration with V. Lakshmibai and C. Musili. In the second edition of the book, conjectures of a standard monomial theory for a general semi-simple (simply-connected) algebraic group, due to Lakshmibai, have been added as an appendix, and the bibliography has been revised.

Standard Monomial Theory

Author : V. Lakshmibai
Publisher : Springer Science & Business Media
ISBN 13 : 3540767576
Total Pages : 271 pages
Book Rating : 4.5/5 (47 download)

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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.

Introduction to the Theory of Standard Monomials

Author : C. S. Seshadri
Publisher : Hindustan Book Agency and Indian National Science Academy
ISBN 13 :
Total Pages : 192 pages
Book Rating : 4.:/5 (321 download)

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Book Synopsis Introduction to the Theory of Standard Monomials by : C. S. Seshadri

Download or read book Introduction to the Theory of Standard Monomials written by C. S. Seshadri and published by Hindustan Book Agency and Indian National Science Academy. This book was released on 2007 with total page 192 pages. Available in PDF, EPUB and Kindle. Book excerpt: "The aim of this book is to give an introduction to what has come to be known as Standard Monomial Theory (SMT). SMT deals with the construction of nice bases of finite dimensional irreducible representations of semi-simple algebraic groups or, in geometric terms, nice bases of coordinate rings of flag varieties (and their Schubert subvarieties) associated to these groups. The book is a reproduction of a course of Lectures given by the author in 1983-84 which appeared in the Brandeis Lecture Notes series."--BOOK JACKET.

Standard Monomial Theory for Reductive Dual Pairs

Author : Steven Glenn Jackson
Publisher :
ISBN 13 :
Total Pages : 204 pages
Book Rating : 4.:/5 (546 download)

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Book Synopsis Standard Monomial Theory for Reductive Dual Pairs by : Steven Glenn Jackson

Download or read book Standard Monomial Theory for Reductive Dual Pairs written by Steven Glenn Jackson and published by . This book was released on 2003 with total page 204 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Flag Varieties

Author : V Lakshmibai
Publisher : Springer
ISBN 13 : 9811313938
Total Pages : 315 pages
Book Rating : 4.8/5 (113 download)

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Book Synopsis Flag Varieties by : V Lakshmibai

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 315 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.

Standard Monomial Theory for Flag Algebras

Author : Sangjib Kim
Publisher :
ISBN 13 :
Total Pages : 112 pages
Book Rating : 4.:/5 (614 download)

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Book Synopsis Standard Monomial Theory for Flag Algebras by : Sangjib Kim

Download or read book Standard Monomial Theory for Flag Algebras written by Sangjib Kim and published by . This book was released on 2005 with total page 112 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Singular Loci of Schubert Varieties

Author : Sara Sarason
Publisher : Springer Science & Business Media
ISBN 13 : 146121324X
Total Pages : 254 pages
Book Rating : 4.4/5 (612 download)

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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.

Representation Theories and Algebraic Geometry

Author : A. Broer
Publisher : Springer Science & Business Media
ISBN 13 : 9401591318
Total Pages : 455 pages
Book Rating : 4.4/5 (15 download)

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Book Synopsis Representation Theories and Algebraic Geometry by : A. Broer

Download or read book Representation Theories and Algebraic Geometry written by A. Broer and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 455 pages. Available in PDF, EPUB and Kindle. Book excerpt: The 12 lectures presented in Representation Theories and Algebraic Geometry focus on the very rich and powerful interplay between algebraic geometry and the representation theories of various modern mathematical structures, such as reductive groups, quantum groups, Hecke algebras, restricted Lie algebras, and their companions. This interplay has been extensively exploited during recent years, resulting in great progress in these representation theories. Conversely, a great stimulus has been given to the development of such geometric theories as D-modules, perverse sheafs and equivariant intersection cohomology. The range of topics covered is wide, from equivariant Chow groups, decomposition classes and Schubert varieties, multiplicity free actions, convolution algebras, standard monomial theory, and canonical bases, to annihilators of quantum Verma modules, modular representation theory of Lie algebras and combinatorics of representation categories of Harish-Chandra modules.

Determinantal Rings

Author : Winfried Bruns
Publisher : Springer
ISBN 13 : 3540392742
Total Pages : 246 pages
Book Rating : 4.5/5 (43 download)

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Book Synopsis Determinantal Rings by : Winfried Bruns

Download or read book Determinantal Rings written by Winfried Bruns and published by Springer. This book was released on 2006-11-14 with total page 246 pages. Available in PDF, EPUB and Kindle. Book excerpt: Determinantal rings and varieties have been a central topic of commutative algebra and algebraic geometry. Their study has attracted many prominent researchers and has motivated the creation of theories which may now be considered part of general commutative ring theory. The book gives a first coherent treatment of the structure of determinantal rings. The main approach is via the theory of algebras with straightening law. This approach suggest (and is simplified by) the simultaneous treatment of the Schubert subvarieties of Grassmannian. Other methods have not been neglected, however. Principal radical systems are discussed in detail, and one section is devoted to each of invariant and representation theory. While the book is primarily a research monograph, it serves also as a reference source and the reader requires only the basics of commutative algebra together with some supplementary material found in the appendix. The text may be useful for seminars following a course in commutative ring theory since a vast number of notions, results, and techniques can be illustrated significantly by applying them to determinantal rings.

The Grassmannian Variety

Author : V. Lakshmibai
Publisher : Springer
ISBN 13 : 1493930826
Total Pages : 174 pages
Book Rating : 4.4/5 (939 download)

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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 174 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.

Computational Algebraic Geometry

Author : Frederic Eyssette
Publisher : Springer Science & Business Media
ISBN 13 : 1461227526
Total Pages : 334 pages
Book Rating : 4.4/5 (612 download)

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Book Synopsis Computational Algebraic Geometry by : Frederic Eyssette

Download or read book Computational Algebraic Geometry written by Frederic Eyssette and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 334 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory and practice of computation in algebraic geometry and related domains, from a mathematical point of view, has generated an increasing interest both for its rich theoretical possibilities and its usefulness in applications in science and engineering. In fact, it is one of the master keys for future significant improvement of the computer algebra systems (e.g., Reduce, Macsyma, Maple, Mathematica, Axiom, Macaulay, etc.) that have become such useful tools for many scientists in a variety of disciplines. The major themes covered in this volume, arising from papers p- sented at the conference MEGA-92 were: - Effective methods and complexity issues in commutative algebra, projective geometry, real geometry, and algebraic number theory - Algebra-geometric methods in algebraic computing and applica tions. MEGA-92 was the second of a new series of European conferences on the general theme of Effective Methods in Algebraic Geometry. It was held in Nice, France, on April 21-25, 1992 and built on the themes presented at MEGA-90 (Livomo, Italy, April 17-21, 1990). The next conference - MEGA-94 - will be held in Santander, Spain in the spring of 1994. The Organizing committee that initiatiod and supervises this bi enniel conference consists of A. Conte (Torino), J.H. Davenport (Bath), A. Galligo (Nice), D. Yu. Grigoriev (Petersburg), J. Heintz (Buenos Aires), W. Lassner (Leipzig), D. Lazard (paris), H.M. MOller (Hagen), T. Mora (Genova), M. Pohst (DUsseldort), T. Recio (Santander), J.J.

Kac-Moody Groups, their Flag Varieties and Representation Theory

Author : Shrawan Kumar
Publisher : Springer Science & Business Media
ISBN 13 : 9780817642273
Total Pages : 630 pages
Book Rating : 4.6/5 (422 download)

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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 2002-09-10 with total page 630 pages. Available in PDF, EPUB and Kindle. Book excerpt: "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

Monomial Algebras

Author : Rafael Villarreal
Publisher : CRC Press
ISBN 13 : 148223470X
Total Pages : 704 pages
Book Rating : 4.4/5 (822 download)

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Book Synopsis Monomial Algebras by : Rafael Villarreal

Download or read book Monomial Algebras written by Rafael Villarreal and published by CRC Press. This book was released on 2018-10-08 with total page 704 pages. Available in PDF, EPUB and Kindle. Book excerpt: Monomial Algebras, Second Edition presents algebraic, combinatorial, and computational methods for studying monomial algebras and their ideals, including Stanley–Reisner rings, monomial subrings, Ehrhart rings, and blowup algebras. It emphasizes square-free monomials and the corresponding graphs, clutters, or hypergraphs. New to the Second Edition Four new chapters that focus on the algebraic properties of blowup algebras in combinatorial optimization problems of clutters and hypergraphs Two new chapters that explore the algebraic and combinatorial properties of the edge ideal of clutters and hypergraphs Full revisions of existing chapters to provide an up-to-date account of the subject Bringing together several areas of pure and applied mathematics, this book shows how monomial algebras are related to polyhedral geometry, combinatorial optimization, and combinatorics of hypergraphs. It directly links the algebraic properties of monomial algebras to combinatorial structures (such as simplicial complexes, posets, digraphs, graphs, and clutters) and linear optimization problems.

Representations of Algebraic Groups

Author : Jens Carsten Jantzen
Publisher : American Mathematical Soc.
ISBN 13 : 082184377X
Total Pages : 594 pages
Book Rating : 4.8/5 (218 download)

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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.

Lectures on Invariant Theory

Author : Igor Dolgachev
Publisher : Cambridge University Press
ISBN 13 : 9780521525480
Total Pages : 244 pages
Book Rating : 4.5/5 (254 download)

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Book Synopsis Lectures on Invariant Theory by : Igor Dolgachev

Download or read book Lectures on Invariant Theory written by Igor Dolgachev and published by Cambridge University Press. This book was released on 2003-08-07 with total page 244 pages. Available in PDF, EPUB and Kindle. Book excerpt: The primary goal of this 2003 book is to give a brief introduction to the main ideas of algebraic and geometric invariant theory. It assumes only a minimal background in algebraic geometry, algebra and representation theory. Topics covered include the symbolic method for computation of invariants on the space of homogeneous forms, the problem of finite-generatedness of the algebra of invariants, the theory of covariants and constructions of categorical and geometric quotients. Throughout, the emphasis is on concrete examples which originate in classical algebraic geometry. Based on lectures given at University of Michigan, Harvard University and Seoul National University, the book is written in an accessible style and contains many examples and exercises. A novel feature of the book is a discussion of possible linearizations of actions and the variation of quotients under the change of linearization. Also includes the construction of toric varieties as torus quotients of affine spaces.

Representation Theory, Number Theory, and Invariant Theory

Author : Jim Cogdell
Publisher : Birkhäuser
ISBN 13 : 3319597280
Total Pages : 630 pages
Book Rating : 4.3/5 (195 download)

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Book Synopsis Representation Theory, Number Theory, and Invariant Theory by : Jim Cogdell

Download or read book Representation Theory, Number Theory, and Invariant Theory written by Jim Cogdell and published by Birkhäuser. This book was released on 2017-10-19 with total page 630 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains selected papers based on talks given at the "Representation Theory, Number Theory, and Invariant Theory" conference held at Yale University from June 1 to June 5, 2015. The meeting and this resulting volume are in honor of Professor Roger Howe, on the occasion of his 70th birthday, whose work and insights have been deeply influential in the development of these fields. The speakers who contributed to this work include Roger Howe's doctoral students, Roger Howe himself, and other world renowned mathematicians. Topics covered include automorphic forms, invariant theory, representation theory of reductive groups over local fields, and related subjects.

Toric Varieties

Author : David A. Cox
Publisher : American Mathematical Society
ISBN 13 : 147047820X
Total Pages : 870 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Toric Varieties by : David A. Cox

Download or read book Toric Varieties written by David A. Cox and published by American Mathematical Society. This book was released on 2024-06-25 with total page 870 pages. Available in PDF, EPUB and Kindle. Book excerpt: Toric varieties form a beautiful and accessible part of modern algebraic geometry. This book covers the standard topics in toric geometry; a novel feature is that each of the first nine chapters contains an introductory section on the necessary background material in algebraic geometry. Other topics covered include quotient constructions, vanishing theorems, equivariant cohomology, GIT quotients, the secondary fan, and the minimal model program for toric varieties. The subject lends itself to rich examples reflected in the 134 illustrations included in the text. The book also explores connections with commutative algebra and polyhedral geometry, treating both polytopes and their unbounded cousins, polyhedra. There are appendices on the history of toric varieties and the computational tools available to investigate nontrivial examples in toric geometry. Readers of this book should be familiar with the material covered in basic graduate courses in algebra and topology, and to a somewhat lesser degree, complex analysis. In addition, the authors assume that the reader has had some previous experience with algebraic geometry at an advanced undergraduate level. The book will be a useful reference for graduate students and researchers who are interested in algebraic geometry, polyhedral geometry, and toric varieties.