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.

Table of Contents

• Introduction
• Subgroups of finite Abelian groups
• Hall-Littlewood symmetric functions
• Some enumerative combinatorics
• Some algebraic combinatorics