*(English)*Zbl 0878.14034

Young tableaux are well studied in representation theory and combinatorics [see the classic books: *G. James* and *A. Kerber*, “The representation theory of the symmetric group” (1981; Zbl 0491.20010) or *G. E. Andrews*, “The theory of partition” (1976; Zbl 0371.10001)]. The well-known author gives the definitions and main theorems in these fields. But his main emphasis is on the application of Young tableaux in (algebraic) geometry. Defining equations for Grassmannians and flag varieties are given. We find Schubert varieties in flag manifolds, Chern classes are introduced, and the geometry of flag varieties is used to construct the Schubert polynomials of Lascoux and Schützenberger. The basic facts about intersection theory on Grassmannians are given, proofs at some times are delegated to the standard books of *R. Hartshorne* [“Algebraic geometry” (3rd edition 1983; Zbl 0531.14001)] and *I. R. Shafarevich* [“Basic algebraic geometry. I and II” (2nd edition 1994; Zbl 0797.14001 and Zbl 0797.14002)] on algebraic geometry. The wealth of presented material makes this excusable.

There are numerous exercises with answers in each chapter and a lot of references to be found at the end of the book.

##### MSC:

14M15 | Grassmannians, Schubert varieties, flag manifolds |

05E10 | Combinatorial aspects of representation theory |

14-02 | Research monographs (algebraic geometry) |

20-02 | Research monographs (group theory) |

05-02 | Research monographs (combinatorics) |

20C30 | Representations of finite symmetric groups |

20C33 | Representations of finite groups of Lie type |

22E45 | Analytic representations of Lie and linear algebraic groups over real fields |