Let L be the lattice of Young diagrams ordered by inclusion and let L_n denote the nth rank, i.e. the Young diagrams of size n. Say that lambda > 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_{mu > lambda} mu

D lambda = sum_{lambda > mu} 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?

Let L be the lattice of Young diagrams ordered by inclusion and let L_n denote the nth rank, i.e. the Young diagrams of size n. Say that lambda > 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_{mu > lambda} mu

D lambda = sum_{lambda > mu} 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?

Edit: Maybe I should ask a more precise question, based on the comment below. As I understand it, in the language of Coxeter groups the symmetric groups are "type A," so the Weyl algebra can be thought of as being associated to type A phenomena. What's the analogue of the Weyl algebra for types B, C, etc.?