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基本説明
This book describes combinatorics involving Young tableaux and some of their uses in representation theory and their uses in representation theory and algbraic geometry.
Full Description
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.
Contents
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; Appendix B.
