This work presents foundational research on two approaches to studying subgroup lattices of finite abelian p-groups. The first approach is linear algebraic in nature and generalizes Knuth's study of subspace lattices. This approach yields a combinatorial interpretation of the Betti polynomials of these Cohen-Macaulay posets. The second approach, which employs Hall-Littlewood symmetric functions, exploits properties of Kostka polynomials to obtain enumerative results such as rank-unimodality. Butler completes Lascoux and Schützenberger's proof that Kostka polynomials are nonnegative, then discusses their monotonicity result and a conjecture on Macdonald's two-variable Kostka functions.
Readership: Research mathematicians.
Author(s): Lynne M. Butler
Series: Memoirs of the American Mathematical Society 539
Publisher: American Mathematical Society
Year: 1994
Language: English
Commentary: Single Pages
Pages: C, vi, 160, B
Introduction
Chapter 1 Subgroups of Finite Abelian Groups
1.1 Methods and Results
1.2 The case \lambda= 1^n: Chains and Invariants
1.3 Motivation: Subgroups and Tabloids
1.4 The case \lambda =/= 1^n: Chains
1.5 The case \lambda=/= 1^n: Invariants
1.6 Birkhoff's standard matrices
1.7 A more natural alternative
Chapter 2 Hall-Littlewood Symmetric Functions
2.1 Introduction
2.2 Littlewood's expression for P(x; q)
2.3 Robinson-Schensted insertion and Jeu de Taquin
2.4 Charge
2.5 Nonnegativity of Kostka polynomials
2.6 Monotonicity of Kostka polynomials
2. 7 Two variable Kostka functions
Appendix A Some enumerative combinatorics
A.1 Partitions and finite abelian groups
Appendix B Some algebraic combinatorics
B.1 Symmetric functions
B.2 Tableau terminology
Bibliography
Back Cover
