Algebraic combinatorics | Lie algebras | Representation theory

Littelmann path model

In mathematics, the Littelmann path model is a combinatorial device due to Peter Littelmann for computing multiplicities without overcounting in the representation theory of symmetrisable Kac–Moody algebras. Its most important application is to complex semisimple Lie algebras or equivalently compact semisimple Lie groups, the case described in this article. Multiplicities in irreducible representations, tensor products and branching rules can be calculated using a coloured directed graph, with labels given by the simple roots of the Lie algebra. Developed as a bridge between the theory of crystal bases arising from the work of Kashiwara and Lusztig on quantum groups and the standard monomial theory of C. S. Seshadri and Lakshmibai, Littelmann's path model associates to each irreducible representation a rational vector space with basis given by paths from the origin to a weight as well as a pair of root operators acting on paths for each simple root. This gives a direct way of recovering the algebraic and combinatorial structures previously discovered by Kashiwara and Lusztig using quantum groups. (Wikipedia).

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From playlist Algebra

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From playlist Gaussian Integral

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From playlist Introduction to Vectors

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From playlist Introduction to Vectors

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Related pages

Weyl character formula | Quantum group | Littlewood–Richardson rule | Levi decomposition | Schubert variety | Monomial representation | Root system | Combinatorics | Hans Freudenthal | General linear group | Representation theory | Graph theory | Kac–Moody algebra | C. S. Seshadri | Mathematics | Weyl group | Young tableau | Cartan subalgebra | Special linear group | Standard monomial theory | Hermann Weyl | Universal enveloping algebra | Issai Schur | Semisimple Lie algebra