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Qiaochu Yuan
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Young's lattice and the Weyl algebra

Let L be the lattice of Young diagrams ordered by inclusion and let Ln 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 = summu > lambda mu

D lambda = sumlambda > 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?

Qiaochu Yuan
  • 118.2k
  • 40
  • 447
  • 741