On the Combinatorics of Crystal Graphs, I
نویسنده
چکیده
In this paper, we continue the development of a new combinatorial model for the characters of the irreducible representations of a complex semisimple Lie group. This model, which was initiated by the author and A. Postnikov, will be referred to as the alcove path model, and can be viewed as a discrete counterpart to the Littelmann path model. Compared to similar models in the literature, the alcove path models leads to a more extensive generalization of the combinatorics in type A (based on Young tableaux, for instance) to arbitrary root systems. The main results of this paper are: (1) a combinatorial description of the crystal graphs corresponding to the irreducible representations; (2) a combinatorial realization of a certain fundamental involution on the canonical basis, which exhibits the crystals as self-dual posets, corresponds to the action of the longest Weyl group element on an irreducible representation, and generalizes Schützenberger’s involution on tableaux; (3) a generalization to arbitrary root systems of Schützenberger’s sliding algorithm (also known as jeu de taquin), which has many applications to the representation theory of the Lie algebra of type A.
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