DIRECT SUM DECOMPOSITIONS OF MATROIDS AND EXPONENTIAL STRUCTURES

We associate to a simple matroid (resp. a geometric lattice) M and a n umber d dividing the rank of M a partially ordered set D-d(M) whose up per intervals are (set-) partition lattices. Indeed, for some importan t cases they are exponential structures in the sense of Stanley [11]. Our construction includes the partition lattice, the poset of partitio ns whose size is divisible by a fixed number d, and the poset of direc t sum decompositions of a finite vector space. If M is a modularly com plemented matroid the posets D-d(M) are CL-shellable. This generalizes results of Sagan and Wachs and settles the open problem of the shella bility of the poset of direct sum decompositions. We analyse the shell ing and derive some facts about the descending chains. We can apply th ese techniques to retrieve the results of Wachs about descending chain s in the lattice of d-divisible partitions. We also derive a formula f or the Mobius number of the poset of direct sum decompositions of a ve ctor space. (C) 1995 Academic Press, Inc.