MEMOIRS OF THE AMERICAN-MATHEMATICAL-SOCIETY - SUBGROUP LATTICES AND SYMMETRICAL FUNCTIONS - INTRODUCTION

This memoir presents foundational research on two approaches used to s tudy the lattice of subgroups of a finite abelian p-group of type lamb da. Such a p-group is isomorphic to Z/p(lambda 1)Z x...x Z/p(lambda l) Z. The first approach, which is linear algebraic in nature and general izes Knuth's study of subspace lattices, establishes this lattice as a n enumerative p-analogue of the product of chains whose lengths are th e parts lambda(i). In particular, we obtain combinatorial descriptions of polynomials that count chains of subgroups and Betti polynomials. Subsequent work, which establishes this subgroup lattice as an order-t heoretic analogue of the chain product and provides a topological expl anation of the nonnegativity of its Betti polynomials, is based on the research in this memoir. The second approach, which employs Hall-Litt lewood symmetric functions, exploits properties of Kostka polynomials to obtain enumerative results for this lattice. In particular, the non negativity of Kostka polynomials, Lascoux and Schutzenberger's proof o f which is completed in this memoir, implies the lattice of subgroups of any finite abelian group is rank-unimodal. The foundational work in this memoir is from the author's PhD thesis, supervised by Richard St anley at MIT and completed in May 1986. Since that time properties of Kostka polynomials other than their nonnegativity have proved useful i n the study of subgroup lattices. In this memoir we also discuss the m onotonicity property, as deduced by Lascoux and Schutzenberger from th eir combinatorial description of Kostka polynomials. Finally we presen t a conjecture on Macdonald's two variable Kostka functions.