Symmetric Functions And Hall Polynomials

Symmetric Functions and Hall Polynomials

This 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, for example, 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 two parameters. 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 symmetric functions, 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 and group-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 of affairs...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 been long 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.

Symmetric Functions and Hall Polynomials

An Introduction to Symmetric Functions and Their Combinatorics

This book is a reader-friendly introduction to the theory of symmetric functions, and it includes fundamental topics such as the monomial, elementary, homogeneous, and Schur function bases; the skew Schur functions; the Jacobi–Trudi identities; the involution ω ω; the Hall inner product; Cauchy's formula; the RSK correspondence and how to implement it with both insertion and growth diagrams; the Pieri rules; the Murnaghan–Nakayama rule; Knuth equivalence; jeu de taquin; and the Littlewood–Richardson rule. The book also includes glimpses of recent developments and active areas of research, including Grothendieck polynomials, dual stable Grothendieck polynomials, Stanley's chromatic symmetric function, and Stanley's chromatic tree conjecture. Written in a conversational style, the book contains many motivating and illustrative examples. Whenever possible it takes a combinatorial approach, using bijections, involutions, and combinatorial ideas to prove algebraic results. The prerequisites for this book are minimal—familiarity with linear algebra, partitions, and generating functions is all one needs to get started. This makes the book accessible to a wide array of undergraduates interested in combinatorics.