On the combinatorics of crystal graphs. I: Lusztig’s involution. (English) Zbl 1129.05058
[For Part II see Proc. Am. Math. Soc. 136, No. 3, 825–837 (2008; Zbl 1129.05058 below).]
In two recent papers the author of the paper under review and A. Postnikov have developed a new combinatorial model for the irreducible characters of a complex semisimple Lie group \(G\); see e.g. C. Lenart and A. Postnikov [Affine Weyl groups in K-theory and representation theory, Int. Math. Res. Not. 2007, No. 12, Article ID rnm038, 65 p. (2007; Zbl 1137.14037)]. This model is called the alcove path model and combines the context of the equivariant K-theory of the generalized flag varity \(G/B\) and ideas based on the Stembridge combinatorial model for Weyl characters. It leads to an extensive generalization of the combinatorics of irreducible characters from Lie type A to arbitrary type. In the present paper the author continues the development of the new model. The approach is type-independent. The main results are: (1) A combinatorial description of the crystal graphs corresponding to the irreducible representations. In particular, the author gives a transparent proof, based on the Yang–Baxter equation, that the mentioned description does not depend on the choice involved in the model; (2) A combinatorial realization of Lusztig’s involution on the canonical basis exhibiting the crystals as self-dual posets; (3) An analog for arbitrary root systems, based on the Yang–Baxter equation, of Schützenberger’s jeu de taquin.
In two recent papers the author of the paper under review and A. Postnikov have developed a new combinatorial model for the irreducible characters of a complex semisimple Lie group \(G\); see e.g. C. Lenart and A. Postnikov [Affine Weyl groups in K-theory and representation theory, Int. Math. Res. Not. 2007, No. 12, Article ID rnm038, 65 p. (2007; Zbl 1137.14037)]. This model is called the alcove path model and combines the context of the equivariant K-theory of the generalized flag varity \(G/B\) and ideas based on the Stembridge combinatorial model for Weyl characters. It leads to an extensive generalization of the combinatorics of irreducible characters from Lie type A to arbitrary type. In the present paper the author continues the development of the new model. The approach is type-independent. The main results are: (1) A combinatorial description of the crystal graphs corresponding to the irreducible representations. In particular, the author gives a transparent proof, based on the Yang–Baxter equation, that the mentioned description does not depend on the choice involved in the model; (2) A combinatorial realization of Lusztig’s involution on the canonical basis exhibiting the crystals as self-dual posets; (3) An analog for arbitrary root systems, based on the Yang–Baxter equation, of Schützenberger’s jeu de taquin.
Reviewer: Vesselin Drensky (Sofia)
MSC:
| 05E15 | Combinatorial aspects of groups and algebras (MSC2010) |
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
| 20G05 | Representation theory for linear algebraic groups |
| 22E46 | Semisimple Lie groups and their representations |
Keywords:
Weyl group; Bruhat order; crystals; canonical basis; Littelmann path model; root operators; Lusztig’s involution; evacuation; Jeu de taquin; \(\lambda\)-chains; admissible subsets; Yang–Baxter movesReferences:
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