Beschreibung:
In earlier work the author introduced the notion of an updown category, which can be regarded as a graded poset with multiplicities and automorphisms. An updown category C naturally has associated linear operators U and D on the graded vector space k(Ob C). In Stanley’s differential posets, the commutator [D,U] is a constant multiple of the identity. We consider various commutation conditionsʺ weaker than this: in particular, the weak commutation conditionʺ that every element of Ob C is an eigenvector for [D,U]. We also show how a Hopf algebra structure on k(Ob C) can provide a way of showing that C satisfies the weak commutation condition. We illustrate with various examples, including updown categories of integer partitions, integer compositions, planar rooted trees, and rooted trees.