Synopses & Reviews
This new and much expanded edition of a well-received book remains the only text available on the subject of symmetric functions and Hall polynomials. There are new sections in almost every chapter, and many new examples have been included throughout.
Review
"It serves as proof of the metatheorem that wherever 'natural' structures indexed by partitions arise, the algebra of symmetric functions is nearby." Mathematical Reviews
Table of Contents
I: Symmetric Functions
1.1. Partitions
1.2. The Ring of Symmetric Functions
1.3. Schur Functions
1.4. Orthogonality
1.5. Skew Schur Functions
1.6. Transition Matrices
1.7. The Characters of the Symmetric Groups
1.8. Plethysm
1.9. The Littlewood-Richardson Rule
Appendix A: Polynomial Functions and Polynomial Representations
Appendix B: Characters of Wreath Products
II: Hall Polynomials
2.1. Finite o-modules
2.2. The Hall Algebra
2.3. The LR-sequence of a Submodule
2.4. The Hall Polynomial
Appendix (by A. Zelevinsky): Another Proof of Hall's Theorem
III: Hall-Littlewood Symmetric Functions
3.1. The Symmetric Polynomials R-y
3.2. Hall-Littlewood Functions
3.3. The Hall Algebra Again
3.4. Orthogonality
3.5. Skew Hall-Littlewood Functions
3.6. Transition Matrices
3.7. Green's Polynomials
3.8. Schur's Q-functions
4.1. The Groups L and M Over a Finite Field
4.2. Conjugacy Classes
4.3. Induction from Parabolic Subgroups
4.4. The Characteristic Map
4.5. Construction of the Characters
4.6. The Irreducible Characters
Appendix: Proof of (5.1)
V: The Hecke Ring of GLn Over a Local Field
5.1. Local Fields
5.2. The Hecke Ring H(G,K)
5.3. Spherical Functions
5.4. Hecke Series and Zeta Functions for GLn(E)
5.5. Hecke Series and Zeta Functions for GSp2n(F)
VI: Symmetric Functions With Two Parameters
6.1. Introduction
6.2. Orthogonality
6.3. The Operators Dr/n
6.4. The Symmetric Functions Functions Py(x;q,t)
6.5. Duality
6.6. Pieri Formulas
6.7. The Skew Functions P-y/u, G-y/u
6.8. Integral Forms
6.9. Another Scalar Product
6.10. Jack's Symmetric Functions
VII: Zonal Polynomials
7.1. Gelfand Pairs and Zonal Spherical Functions
7.2. The Gelfand Pair (S2n, Hn)
7.3. The Gelfand Pair [GLn(R),O(n)]
7.4. Integral Formulas
7.5. The Complex Case
7.6. The Quaternionic Case