Séminaire Lotharingien de Combinatoire, B41b (1998), 23pp.

Marc A. A. van Leeuwen

An analogue of Jeu de taquin for Littelmann's crystal paths

Abstract. Littelmann has given a combinatorial model for the characters of representations of semisimple Lie algebras, in terms of certain paths traced in the space of (rational) weights. From it, a description of the decomposition of tensor products can be derived that generalises the Littlewood-Richardson rule (the latter is valid in type A(n) only). We present a new combinatorial construction that expresses in a bijective manner the symmetry of the tensor product in this path model. In type A(n), where there is a correspondence between paths and skew tableaux, this construction is equivalent to Schützenberger's jeu de taquin; in the general case the construction retains its most crucial properties of symmetry and confluence.

Received: August 10, 1998; Accepted: October 5, 1998.

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