## Young Tableaux: With Applications to Representation Theory and GeometryYoung 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. |

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### Contents

Preface page | 8 |

Bumping and sliding | 8 |

Words the plactic monoid | 17 |

Increasing sequences proofs of the claims | 30 |

The RobinsonSchenstedKnuth correspondence | 36 |

The LittlewoodRichardson rule | 58 |

Symmetric polynomials | 72 |

Representations of the symmetric group | 83 |

Representations of the general linear group | 104 |

Flag varieties | 131 |

Schubert varieties and polynomials | 154 |

Appendix A Combinatorial variations | 183 |

Appendix B On the topology of algebraic varieties | 211 |

### Other editions - View all

Young Tableaux: With Applications to Representation Theory and Geometry Mr William Fulton,William Fulton No preview available - 1997 |

Young Tableaux: With Applications to Representation Theory and Geometry Mr William Fulton,William Fulton No preview available - 1997 |

### Common terms and phrases

action algebraic array balls basic basis boxes bumping bundle called canonical Chapter closed column complex conjugate consider consists construction contains coordinate corner Corollary corresponding defined definition denote describe determines dimension dual elementary elements embedding entries equal equation exactly example Exercise fact fixed flag follows formula functions given gives homomorphism ideal identity increasing integers interchanging irreducible isomorphism Knuth equivalent Lemma length letters linear Littlewood-Richardson rule manifold matrix Note obtained occurs pair particular partition permutation polynomials position preceding projective Proof Proposition prove relations representation result reverse ring Schubert Schur seen sequence shape Show skew tableau slide space spanned standard tableau strictly subset subvariety suffices Suppose symmetric takes theorem unique variety vector weight word write zero