Mathematics Colloquia and Seminars

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On any partially ordered set we can consider the naturally defined linear operators "Down", D and "Up", U. In Young's Lattice of partitions ordered by inclusion of diagrams, these operators satisfy the particularly nice relation: DU - UD = I. Enumerative problems concerning Young tableaux (chains in Young's Lattice) can be rephrased in terms of these operators as simple partial differential equations, which can then be solved. On the combinatorial side, many of these results can also be derived from Fomin's pictorial presentation of the Robinson-Schensted correspondence. This approach to Schensted also simplifies the construction of variations and generalizations. Terwilliger studied a natural cubic generalization of this relation which holds for the class of "Uniform posets". This leads to the notion of "Down-up Algebras", where the relation greatly aids the study of highest weight representations.