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Differential Posets, Down-up Algebras, and the Robinson-Schensted-Fomin Machine
Colloquium| Speaker: | Prof. Tom Roby, California State Univ. Hayward. |
| Location: | 693 Kerr |
| Start time: | Mon, Nov 20 2000, 4:10PM |
Description
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.