Commutation relations in Lie theory give rise to families of finite partially ordered sets connected with Kostant's partition function. Techniques originally used to obtain arithmetic properties of Kostant's function mesh well with these poset structures giving a great deal of geometric structure to the Hasse diagrams. Quite apart from any use in Lie theory these posets offer a wealth of examples of nicely behaved tractable posets which are neither rank symmetric nor satisfy LYM.
