Direct sum decompositions of matroids and exponential structures
The author associates to a simple matroid (resp. a geometric lattice) \(M\) and a number \(d\) dividing the rank of \(M\) a partially ordered set \({\mathcal D}_d(M)\) whose upper intervals are (set-) partition lattices. Indeed, for some important cases they are exponential structures in the sense of Stanley. This 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 \({\mathcal D}_d(M)\) 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. By analyzing the shelling, the author derives some facts about the descending chains and applies these techniques to retrieve the results of Wachs about descending chains in the lattice of \(d\)-divisible partitions. A formula for the Möbius number of the poset of direct sum decompositions of a vector space is also obtained in the paper.
scientific article; zbMATH DE number 767692