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**Symmetric functions and Hall polynomials.
2nd ed.**
*(English)*
Zbl 0824.05059

Oxford: Clarendon Press. x, 475 p. (1995).

Since its appearance in 1979 (Zbl 0487.20007), the first edition of this book has been the source and reference book for anything involving symmetric functions. In the meantime, interest in important new symmetric functions arose in several different areas, such as zonal polynomials and Jack polynomials. In 1988, Macdonald introduced a family of symmetric polynomials with two parameters, now called ‘Macdonald polynomials’, that contain Schur functions, Hall-Littlewood polynomials, and the aforementioned zonal and Jack polynomials as special cases. Currently there is plenty of research going on on Macdonald polynomials and related topics. Many intriguing problems are still open. Also, in an increasing number of instances it is discovered that Macdonald polynomials play a significant role. The second edition therefore includes two new chapters about these new symmetric functions. Otherwise, the five chapters of the first edition are more or less unchanged, except for a few additions like a section on Schur’s \(Q\)-functions and a significant enlargement of the examples sections. For those who do not know the first edition I recall that the characteristics of Macdonald’s style are to present the basic theory in the text, which is terse, but precise and to the point, and to have a large list of ‘examples’ at the end of each section (very often larger than the section itself) in which the reader finds a host of more information, examples, applications, and advanced results. Evidently, this second edition (which is twice as large as the first edition) will be the source and reference book for symmetric functions in the next future.

Reviewer: Ch.Krattenthaler (Wien)

### MSC:

05E05 | Symmetric functions and generalizations |

05-02 | Research exposition (monographs, survey articles) pertaining to combinatorics |

05E10 | Combinatorial aspects of representation theory |

05E35 | Orthogonal polynomials (combinatorics) (MSC2000) |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

05A17 | Combinatorial aspects of partitions of integers |

05A15 | Exact enumeration problems, generating functions |

20C25 | Projective representations and multipliers |

20C30 | Representations of finite symmetric groups |

### Keywords:

tableaux; Hall polynomials; orthogonal polynomials; Gelfand pairs; finite general linear groups; symmetric functions; zonal polynomials; Jack polynomials; Schur functions; Hall-Littlewood polynomials; Macdonald polynomials; \(Q\)-functions### Citations:

Zbl 0487.20007
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\textit{I. G. Macdonald}, Symmetric functions and Hall polynomials. 2nd ed. Oxford: Clarendon Press (1995; Zbl 0824.05059)

### Digital Library of Mathematical Functions:

§18.37(iii) OP’s Associated with Root Systems ‣ §18.37 Classical OP’s in Two or More Variables ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials§26.19 Mathematical Applications ‣ Applications ‣ Chapter 26 Combinatorial Analysis

§35.4(ii) Properties ‣ §35.4 Partitions and Zonal Polynomials ‣ Properties ‣ Chapter 35 Functions of Matrix Argument

§35.4(i) Definitions ‣ §35.4 Partitions and Zonal Polynomials ‣ Properties ‣ Chapter 35 Functions of Matrix Argument

### Online Encyclopedia of Integer Sequences:

Coefficient of s(2n) in s(n,n) * s(n,n) * s(n,n) * s(n,n) * s(n,n) * s(n,n), where s(n,n) is the Schur function indexed by two parts of size n, s(2n) is the Schur function corresponding to the trivial representation and * represents the inner or Kronecker product.Coefficient of s(2n) in s(n,n) * s(n,n) * s(n,n) * s(n,n) * s(n,n), where s(2n) is the Schur function corresponding to the trivial representation, s(n,n) is a Schur function corresponding two the two row partition and * represents the inner or Kronecker product of symmetric functions.