Young Tableaux: With Applications to Representation Theory and Geometry (London Mathematical Society Student Texts, Series Number 35)

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FULLING Braids and coverings: selected topics, VAGN LUNDSGAARD HANSEN Steps in commutative algebra, R.Y. SHARP Communication theory, CM. GOLDIE & R.G.E. PINCH Representations of finite groups of Lie type, FRANCOIS DIGNE & JEAN MICHEL Designs, graphs, codes, and their links, P.J. CAMERON & J.H. VAN LINT Complex algebraic curves, FRANCES KIRWAN Lectures on elliptic curves, J.W.S. CASSELS Hyperbolic geometry, BIRGERIVERSEN An introduction to the theory of L-functions and Eisenstein series, H. HIDA Hilbert space: compact operators and the trace theorem, J.R. RETHERFORD Potential theory in the complex plane, T. RANSFORD Undergraduate commutative algebra, M. REID Introduction to computer algebra, A. COHEN Complex algebraic surfaces, A. BEAUVILLE Lectures on Lie groups and Lie algebras, R. CARTER, G. SEGAL & I. MACDONALD A primer of algebraic D-modules, S.C COUTINHO Laplacian operators in differential geometry, S. ROSENBERG Young tableaux, W. FULTON Young tableaux are fillings of the boxes of diagrams that correspond to partitions with positive integers, that are strictly increasing down columns and weakly increasing along rows. The aim of this book is to develop the combinatorics of Young tableaux and to show them in action in the algebra of symmetric functions, the representations of the symmetric and general linear groups, and the geometry of flag varieties. Many of these applications have not been available in book form. In the first part of the book the author develops the basic combinatorics of Young tableaux, including the remarkable constructions of "bumping" and "sliding" that can be used to make them into a monoid, and several interesting correspondences. In Part II these results are used to study representations of the symmetric and general linear groups. In Part III we see relations with geometry on Grassmanians and flag manifolds, including their Schubert subvarieties, and the related Schubert polynomials. Two appendices contain variations of the combinatorics of Part I and the topology needed to relate subvarieties to cohomology classes. The combinatorial chapters of the book are self-contained so that students will find the discussion easy to follow. Researchers in representation theory and algebraic geometry as well as in combinatorics will find Young Tableaux interesting and useful. The Aim Of This Book Is To Develop The Combinatorics Of Young Tableaux And To Show Them In Action In The Algebra Of Symmetric Functions, Representations Of The Symmetric And General Linear Groups, And The Geometry Of Flag Varieties. The First Part Of The Book Is A Self-contained Presentation Of The Basic Combinatorics Of Young Tableaux, Including The Remarkable Constructions Of 'bumping' And 'sliding', And Several Interesting Correspondences. In Part Ii These Results Are Used To Study Representations With Geometry On Grassmannians And Flag Manifolds, Including Their Schubert Subvarieties, And The Related Schubert Polynomials. Much Of This Material Has Never Appeared In Book Form.there Are Numerous Exercises Throughout, With Hints Or Answers Provided. Researchers In Representation Theory And Algebraic Geometry As Well As In Combinatorics Will Find Young Tableaux Interesting And Useful; Students Will Find The Intuitive Presentation Easy To Follow. William Fulton. Includes Bibliographical References (p. 248-253) And Indexes. Added pages 46,47 This book develops the combinatorics of Young tableaux and shows them in action in the algebra of symmetric functions, representations of the symmetric and general linear groups, and the geometry of flag varieties. The first part of the book is a self-contained presentation of the basic combinatorics of Young tableaux, including the remarkable constructions of "bumping" and "sliding", and several interesting correspondences. In Part II the author uses these results to study representations with geometry on Grassmannians and flag manifolds, including their Schubert subvarieties, and the related Schubert polynomials. Much of this material has never before appeared in book form. There are numerous exercises throughout, with hints and answers provided. Researchers in representation theory and algebraic geometry as well as in combinatorics will find this book interesting and useful, while students will find the intuitive presentation easy to follow. The aim of this book is to develop the combinatorics of Young tableaux and to show them in action in the algebra of symmetric functions, representations of the symmetric and general linear groups, and the geometry of flag varieties. The first part of the book is a self-contained presentation of the basic combinatorics of Young tableaux, including the remarkable constructions of 'bumping' and 'sliding', and several interesting correspondences. In Part II these results are used to study representations with geometry on Grassmannians and flag manifolds, including their Schubert subvarieties, and the related Schubert polynomials. Much of this material has never appeared in book form. There are numerous exercises throughout, with hints or answers provided. Researchers in representation theory and algebraic geometry as well as in combinatorics will find Young Tableaux interesting and useful; students will find the intuitive presentation easy to follow The first algorithm, called row-insertion or row bumping, takes a tableau T, and a positive integer x, and constructs a new tableau, donated T x.