An Introduction to Symmetric Functions and their Combinatorics
My book
An Introductions to Symmetric Functions and Their Combinatorics was published by the AMS in November of 2019 as volume 91 in the Student Mathematical Library Series. It's aimed at undergraduates who have studied combinatorics, but it's also a good source for graduate students and research mathematicians who want an introduction to the subject with numerous examples that doesn't rely on a substantial amount of background in abstract algebra. Here are the topics the book covers.
- Symmetric Polynomials, Symmetric Functions, the Monomials Symmetric Polynomials and the Monomial Symmetric Functions
- The Elementary, Complete Homogeneous, and Power Sum Symmetric Functions
- Evaluations of Symmetric Functions and Identities for Binomial Coefficients, Stirling Numbers of Both Kinds, and q-Binomial Coefficients
- Schur Polynomials and Schur Functions as Generating Functions for Semistandard Tableaux and as Ratios of Determinants
- Skew Schur Functions, Stable Grothendieck Functions, Dual Stable Grothendieck Functions, and the Chromatic Symmetric Function
- The Jacobi-Trudi Identities via Nonintersecting Lattice Paths
- The Involution ω
- The Hall Inner Product and Cauchy's Formula
- The RSK Correspondence via Insertion and via Growth Diagrams
- The Pieri Rules and the Murnaghan-Nakayama Rule
- The Littlewood Richardson Rule for Products of Schur Functions and for Skew Schur Functions, via Knuth Equivalence and jeu de taquin
Are you using this book for a course you're teaching, or a course you're taking? Please email me! I'd love to know where and how the book is being used. Even if you're not using the book for a course, I would love to get your feedback.
I have compiled a list of corrections, which includes all of the corrections I've received so far.
I am also compiling a list of commentary.
This list is not yet complete, so please check back regularly for more information.