An Introduction to Symmetric Functions and Their Combinatorics

The theory of symmetric functions is deep with interesting connections to combinatorics and the representation theory of the symmetric group.  A relatively large community of algebraic combinatorialists use symmetric functions in their research.

 

Although the basic theory is accessible to undergraduates, the standard references on symmetric functions are written for an advanced audience.  An Introduction to Symmetric Functions and Their Combinatorics fills this gap in the literature.  It is a lucid introduction for both undergraduates and non-specialists who are looking to quickly learn the fundamentals of symmetric functions.

 

Most major beginning topics in the theory are discussed in detail.  The development is conventional, starting by carefully introducing the five standard bases for the symmetric polynomials in a finite number of variables (the monomial, elementary, homogeneous, power, and Schur) before moving on to many of the classic combinatorial algorithms involving symmetric functions (such as the RSK algorithm, the Pieri rules, the Munrnaghan-Nakayama rule, and the Littlewood-Richardson rule).  There are short detours into more specialized topics such as Grothendieck polynomials and the chromatic symmetric function.

 

The text is well-written and the diagrams included in the book are nicely drawn.  A few pages use color to great effect.  There are plenty of exercises, making the book a great choice for a reading course with an undergraduate, a junior/senior-level topics course, or a senior seminar.  This is definitely the book to give to an undergraduate or beginning graduate student interested in symmetric functions.

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Anthony Mendes (aamendes@calpoly.edu) is a mathematics professor at Cal Poly in San Luis Obispo.