A class of geometric lattices based on finite groups

Abstract

For any finite group G and positive integer n a finite geometric lattice Q n (G) of rank n , the lattice of partial G -partitions, is constructed. Let P n+1 be the lattice of partitions of an ( n +1)-set. There exists a surjection π: Q n (G) → P n+1 , and an injection t: P n+1 → Q n+1 (G) , each of which preserves order and rank. When G is the trivial group, π=t −1 reduces to an isomorphism. The interval structure, Möbius function, and characteristic polynomial of Q n (G) are determined, and Stirling-like identities for the Whitney numbers obtained. The existence of a Boolean sublattice of modular elements in Q n (G) is established, implying that Q n (G) is supersolvable. It is further shown that non-isomorphic groups give non-isomorphic lattices, and the representation problem is solved completely: Q n (G) is representable over a field when n ≥3 if and only if G is isomorphic to a subgroup of the multiplicative group of the field. Consequently, Q n (G) is representable over no field iff G is noncyclic, and, if G is cyclic of order m , then Q n (G) is representable over (a) every field iff m =1, (b) a finite field of order q iff m divides q −1, (c) the rational or real fields iff m =1 or 2, and (d) the complex field for all m .