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.
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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.
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12 of 12 people found the following review helpful:
5.0 out of 5 stars
A Fine Synthesis of Combinatorics, Geometry, and Algebra,
By Peter M. Magyar "peter-the-math-geek" (East Lansing, MI United States) - See all my reviews (REAL NAME)
This review is from: Young Tableaux: With Applications to Representation Theory and Geometry (London Mathematical Society Student Texts) (Paperback)
With his usual lucidity, Fulton brings together the surprisingly wide area of mathematics concerned with Young tableaux. These are combinatorial patterns which index basis vectors of group representations (either of the symmetric group or the general linear group). These vectors can be seen as Plucker coordinate functions on non-linear representations, namely homogeneous spaces (Grassmannians and flag varieties). Thus, Young tableaux form an invaluable tool to examine these representations and varieties in concrete detail. Fulton also gives a good exposition of the combinatorial operations on tableaux which reflect the crystal basis structure from quantum GL(n), though Fulton does not explicitly discuss quantum groups. Other good expositions of these topics, from a more algebraic and combinatorial point of view, are Sagan's newly revised "The Symmetric Group", and Stanley's "Enumerative Combinatorics", Vol 2.
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Inside This Book
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First Sentence:
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. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
skew tableaux, lexicographic array, column bumping, bumping algorithm, tableau pair, row bumping, lattice word, hook length formula, skew diagram, whose rectification, same shape changes, standard tableaux, reverse numbering, nonnegative integer entries, alternating representation, row numbering, tautological line bundle, skew shape, symmetry theorem, given tableau, conjugate partition, resulting tableau, flag varieties, additive basis, flag manifold Key Phrases - Capitalized Phrases (CAPs): (learn more)
Row Bumping Lemma, Proof Let New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
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. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
skew tableaux, lexicographic array, column bumping, bumping algorithm, tableau pair, row bumping, lattice word, hook length formula, skew diagram, whose rectification, same shape changes, standard tableaux, reverse numbering, nonnegative integer entries, alternating representation, row numbering, tautological line bundle, skew shape, symmetry theorem, given tableau, conjugate partition, resulting tableau, flag varieties, additive basis, flag manifold Key Phrases - Capitalized Phrases (CAPs): (learn more)
Row Bumping Lemma, Proof Let New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 22
books:
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1
book
cites this book:
- Differential Topology by Victor Guillemin in Back Matter
- Group Representations in Probability and Statistics (Lecture Notes Vol 11) by Persi Diaconis in Back Matter
- Principles of Algebraic Geometry by Phillip Griffiths in Back Matter
- Symmetric Functions and Hall Polynomials (Oxford Mathematical Monographs) by I. G. Macdonald in Back Matter
- Topologies and Uniformities (Springer Undergraduate Mathematics Series) by I.M. James in Back Matter
See all 22 books this book cites
- Physical Combinatorics (Progress in Mathematics) by Masaki Kashiwara on page 138
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