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Zbl 0824.05059
Macdonald, Ian Grant
Symmetric functions and Hall polynomials. 2nd ed. (English)
[B] Oxford: Clarendon Press. x, 475 p. \sterling\ 55.00 (1995). ISBN 0-19-853489-2/hbk

Since its appearance in 1979 (Zbl 0487.20007), the first edition of this book has been {\it 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 {\it the} source and reference book for symmetric functions in the next future.
[Ch.Krattenthaler (Wien)]

MSC 2000:: *05E05 Symmetric functions
05-02 Research monographs (combinatorics)
05E10 Tableaux, etc.
05E35 Orthogonal polynomials (combinatorics)
20-02 Research monographs (group theory)
05A17 Partitions of integres (combinatorics)
05A15 Combinatorial enumeration problems
20C25 Projective representations and multipliers of groups
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

Cited in: Zbl pre05918879 Zbl 1200.05246 Zbl 1168.53044 Zbl 1176.05010 Zbl 1166.16007 Zbl 1106.33015 Zbl 1095.20003 Zbl 1186.05122 Zbl 1085.60014 Zbl 1032.33010 Zbl 0947.33015 Zbl 0937.05008 Zbl 0936.33007 Zbl 0918.22013 Zbl 0939.05090 Zbl 0924.17006 Zbl 0916.05077 Zbl 0899.05068 Zbl 0895.05065 Zbl 0894.05053 Zbl 0876.05098 Zbl 0906.06003 Zbl 0865.05075 Zbl 0864.05085 Zbl 0863.05081 Zbl 0861.05063 Zbl 0858.05098 Zbl 0848.05067 Zbl 0844.05097

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