An Introduction to Symmetric Functions and Their Combinatorics
Eric S. Egge
- English
- PDF
- Available on iOS & Android
An Introduction to Symmetric Functions and Their Combinatorics
Eric S. Egge
About This Book
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 JacobiâTrudi identities; the involution $\omega$; 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 MurnaghanâNakayama rule; Knuth equivalence; jeu de taquin; and the LittlewoodâRichardson 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 minimalâfamiliarity 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.