Young Tableaux With Applications to Representation Theory and Geometry [Paperback]

Describes combinatorics involving Young tableaux and their uses in representation theory and algebraic geometry.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 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.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.Part I. Calculus Of Tableux: 1. Bumping and sliding; 2. Words: the plactic monoid; 3. Increasing sequences: proofs of the claims; 4. The Robinson-Schensted-Knuth Correspondence; 5. The Littlewood-Richardson rule; 6. Symmetric polynomials; Part II. Representation Theory: 7. Representations of the symmetric group; 8. Representations of the general linear group; Part III. Geometry: 9. Flag varieties; 10. Schubert varieties and polynomials; Appendix A; AppendilC