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A plactic algebra for semisimple Lie algebras. (English) Zbl 0892.17009
Let $$G$$ be a simple, simply connected algebraic group, and $$\mathfrak g$$ its Lie algebra. A plactic algebra can be thought of as a (noncommutative) model for the representation ring of $$\mathfrak g$$, and was introduced in A. Lascoux and M. P. Schützenberger [Le monoide plaxique, in ‘Noncommutative structures in algebra and geometric combinatorics’, Quad. Ric. Sci. 109, 129-156 (1981; Zbl 0517.20036)] and M. P. Schützenberger [La correspondance de Robinson, in ‘Combinatoire de répresentation du groupe symetrique’, Strasbourg 1976, Lect. Notes Math. 579, 59-113 (1977; Zbl 0398.05011)].
The author considers the case of the plactic algebra $${\mathbb ZP}$$ of equivalence classes of paths in the space of rational weights; see P. Littelmann [Ann. Math., II. Ser. 142, 499-525 (1995; Zbl 0858.17023), Proc. Int. Congr. Math., Zürich 1994, Vol. 1, 298-308 (1995; Zbl 0848.17021)] and A. Joseph [Quantum groups and their primitive ideals (Springer-Verlag, New York) (1994; Zbl 0808.17004)].
In the paper under review, the author provides a description of this algebra more in the spirit of Lascoux and Schützenberger. Let $$V$$ be a faithful representation and let $$\mathbb D$$ be the associated set of Lakshmibai-Seshadri paths; this is a basis for the corresponding path model of $$V$$. Let $${\mathbb Z}\{{\mathbb D}\}$$ be the free associative algebra on $$\mathbb D$$. The canonical projection from $${\mathbb Z}\{{\mathbb D}\}$$ to $${\mathbb ZP}$$, defined on monomials by concatenation of paths, is surjective; the author proves several theorems giving explicit descriptions of the kernel of this map, thus providing descriptions of the plactic algebra. To do this, he also develops a combinatorial description of standard monomials of paths using ideas from standard monomial theory [see, for example, V. Lakshmibai and C. S. Seshadri, Standard monomial theory, in ‘Proceedings of the Hyderabad conference on algebraic groups’, 1989, 279-322 (1991; Zbl 0785.14028)] and also obtains a Demazure character formula in terms of standard monomials. In particular, he shows that in type $$A_n$$, $$\mathbb ZP$$ is the plactic algebra defined by Lascoux and Schützenberger.

##### MSC:
 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 05E10 Combinatorial aspects of representation theory 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
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