Let L be the lattice of Young diagrams ordered by inclusion and let L<sub>n</sub> denote the nth rank, i.e. the Young diagrams of size n.  Say that lambda &gt; mu if lambda covers mu, i.e. mu can be obtained from lambda by removing one box and let C[L] be the free vector space on L.  The operators

U lambda = sum<sub>mu &gt; lambda</sub> mu

D lambda = sum<sub>lambda &gt; mu</sub> mu

are a decategorification of the induction and restriction operators on the symmetric groups, and (as observed by Stanley and generalized in the theory of differential posets) they have the property that DU - UD = I; in other words, Young's lattice along with U, D form a representation of the Weyl algebra. 

Is this a manifestation of a more general phenomenon?  What's the relationship between differential operators and the representation theory of the symmetric group?