Group Characters, Symmetric Functions, and the Hecke Algebra (University Lecture Series)

Directed at graduate students and mathematicians, this book covers an unusual set of interrelated topics, presenting a self-contained exposition of the algebra behind the Jones polynomial along with various excursions into related areas. The book is made up of lecture notes from a course taught by Goldschmidt at the University of California at Berkeley in 1989. The course was organized in three parts. Part I covers, among other things, Burnside's Theorem that groups of order $p^aq^b$ are solvable, Frobenius' Theorem on the existence of Frobenius kernels, and Brauer's characterization of characters. Part II covers the classical character theory of the symmetric group and includes an algorithm for computing the character table of $S^n$ ; a construction of the Specht modules; the "determinant form" for the irreducible characters; the hook-length formula of Frame, Robinson, and Thrall; and the Murnaghan-Nakayama formula. Part III covers the ordinary representation theory of the Hecke algebra, the construction of the two-variable Jones polynomial, and a derivation of Ocneanu's "weights" due to T. A. Springer. Title 1 Copyright 2 Contents 3 Preface 4 Part I 5 Chapter 1. Finite-Dimensional Algebras 6 Chapter 2. Group Characters 10 Chapter 3. Divisibility 15 Chapter 4. Induced Characters 18 Chapter 5. Further Results 22 Part II 27 Chapter 6. Permutations and Partitions 28 Chapter 7. The Irreducible Characters of $S^n$ 33 Chapter 8. The Specht Modules 36 Chapter 9. Symmetric Functions 41 Chapter 10. The Schur Functions 45 Chapter 11. The Littlewood-Richardson Ring 49 Chapter 12. Two Useful Formulas 53 Part III 57 Chapter 13. The Hecke Algebra 58 Chapter 14. The Markov Trace 63 Bibliography 68 Presents an exposition of the algebra behind the Jones polynomial along with various excursions into related areas. This book covers Burnside's Theorem that groups of order $p^aq^b$ are solvable, Frobenius' Theorem on the existence of Frobenius kernels, and Brauer's characterization of characters.