Generalizing the Combinatorics of Young Tableaux to Arbitrary Lie Type

   Cristian Lenart (SUNY Albany)

 Young tableaux provide a combinatorial model for the irreducible characters of the Lie algebra of type $A$. Their very rich combinatorics has been widely studied, especially since the pioneering work of Lascoux and Sch\"utzenberger in the 1970s and 1980s. We present a simple combinatorial model for the irreducible characters of an arbitrary semisimple Lie algebra, which allows us to generalize much of the combinatorics of Young tableaux. More precisely, among the features of our model are: (1) a Littlewood-Richardson rule (for decomposing the tensor product of irreducible representations); (2) root operators (which give rise to the corresponding crystal graph structure, and generalize the coplactic operations on tableaux); (3) a generalization of the left and right keys of tableaux (related to Demazure characters);  (4) an explicit combinatorial description (generalizing Sch\"utzenberger's ``evacuation'' procedure for tableaux) of the action of the longest Weyl group element on canonical bases. Our model has certain advantages over Littelmann paths; for instance, no explicit description of the later involution is known in Littelmann's model. This is a joint work with Alexander Postnikov.

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