Direct Sum Decompositions of Matroids and Exponential Structures

We associate to a simple matroid (resp. a geometric lattice) M and a number d dividing the rank of M a partially ordered set Dd(M) whose upper intervals are (set-) partition lattices. Indeed, for some important cases they are exponential structures in the sense of Stanley [11]. Our construction includes the partition lattice, the poset of partitions whose size is divisible by a fixed number d, and the poset of direct sum decompositions of a finite vector space. If M is a modularly complemented matroid the posets Dd(M) are CL-shellable. This generalizes results of Sagan and Wachs and settles the open problem of the shellability of the poset of direct sum decompositions. We analyse the shelling and derive some facts about the descending chains. We can apply these techniques to retrieve the results of Wachs about descending chains in the lattice of d-divisible partitions. We also derive a formula for the Mِbius number of the poset of direct sum decompositions of a vector space.