## Symmetric Functions and Hall PolynomialsThis is a paperback version of the second, much expanded, edition of Professor Macdonald's acclaimed monograph on symmetric functions and Hall polynomials. Almost every chapter has new sections and new examples have been included throughout. Extra material in the appendix to Chapter 1, forexample, includes an account of the related theory of polynomial representations of the general linear groups (always in characteristic zero). Chapters 6 and 7 are new to the second edition: Chapter 6 contains an extended account of a family of symmetric functions depending rationally on twoparameters. These symmetric functions include as particular cases many of those encountered earlier in the book but they also include, as a limiting case, Jack's symmetric functions depending on a parameter (. Many of the properties of the Schur functions generalize to these two-parameter symmetricfunctions, but the proofs (at present) are usually more elaborate. Chapter 7 is devoted to the study of the zonal polynomials, long familiar to statisticians. From one point of view they are a special case of Jack's symmetric functions (the parameter ( being equal to 2) but their combinatorial andgroup-theoretic connections make them worthy of study in their own right. From reviews of the first edition: 'Despite the amount of material of such great potential interest to mathematicians...the theory of symmetric functions remains all but unknown to the persons it is most likely to benefit...Hopefully this beautifully written book will put an end to this state ofaffairs...I have no doubt that this book will become the definitive reference on symmetric functions and their applications.' Bulletin of the AMS '...In addition to providing a self-contained and coherent account of well-known and classical work, there is a great deal which is original. The book is dotted with gems, both old and new...It is a substantial and valuable volume and will be regarded as the authoritative source which has beenlong awaited in this subject.' LMS book reviews From reviews of the second edition: 'Evidently this second edition will be the source and reference book for symmetric functions in the near future.'Zbl. Math. |

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### 內容

SYMMETRIC FUNCTIONS | 1 |

1 1 Example | 8 |

HALL POLYNOMIALS | 179 |

HALLLITTLEWOOD SYMMETRIC | 204 |

The Hall algebra again | 215 |

Orthogonality | 222 |

Transition matrices | 238 |

Greens polynomials | 246 |

SYMMETRIC FUNCTIONS WITH | 305 |

The operators Dn | 317 |

Duality | 327 |

The skew functions P QVM | 343 |

Integral forms | 352 |

Another scalar product | 368 |

Jacks symmetric functions | 376 |

ZONAL POLYNOMIALS | 387 |

Schurs functions 250 | 265 |

Induction from parabolic subgroups | 273 |

Construction of the characters | 280 |

THE HECKE RING OF GLn OVER | 292 |

Spherical functions | 298 |

457 | |

NOTATION | 468 |

474 | |

### 常見字詞

algebraically independent basis border strip Chapter coefficient column column-strict commutative conjugacy classes conjugate corresponding cotype cycle-type Deduce defined definition denote the number determinant double coset eigenvalues equal equivalent Example 9 finite o-module follows form a Z-basis formula Frobenius function on G functor Gelfand pair Hall-Littlewood hence homogeneous of degree homomorphism hook-lengths horizontal strip identity induction integer involution irreducible characters isomorphism Let G linear combination mapping module monomial non-negative integers notation Notes and references obtain operator orthogonality p-core P,oof particular partition of length permutation plane partitions polynomial function positive integers proof representation resp right-hand side ring scalar product Schur functions sequence Show skew diagram square standard tableaux strict partition strictly upper strip of length subgroup submodule subsets subspace summed over partitions symmetric group tableau tableaux of shape unitriangular variables vector space x e G zero zonal polynomials zonal spherical functions