Littelmann path model: Difference between revisions

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In [[mathematics]], the '''Littelmann path model''' is a [[combinatorics|combinatorial device]] due to [[Peter Littelmann]] for computing multiplicities ''without overcounting'' in the [[representation theory]] of symmetrisable [[Kac-Moody algebra]]s. Its most important application is to complex [[semisimple Lie algebra]]s or equivalently compact [[semisimple Lie group]]s, the case described in this article. Multiplicities in [[Weyl character formula|irreducible representations]], tensor products and [[branching rule]]s can be calculated using a [[graph theory|coloured directed graph]], with labels given by the [[root system|simple roots]] of the Lie algebra.

Developed as a bridge between the theory of [[crystal basis|crystal bases]] arising from the work of [[Masaki Kashiwara|Kashiwara]] and [[George Lusztig|Lusztig]] on [[quantum group]]s 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 [[root lattice|weight]] as well as a pair of '''root operators''' acting on paths for each [[root lattice|simple root]]. This gives a direct way of recovering the algebraic and combinatorial structures previously discovered by Kashiwara and Lustzig using quantum groups.

==Background and motivation==

Some of the basic questions in the representation theory of complex semisimple Lie algebras or compact semisimple Lie groups going back to [[Hermann Weyl]] include:<ref>{{harvnb|Weyl|1946}}</ref><ref>{{harvnb|Humphreys|1994}}</ref>

* For a given [[dominant weight]] '''&lambda;λ''', find the weight multiplicities in the [[Weyl character formula|irreducible representation]] ''L''(λ) with highest weight λ.

* For two highest weights λ, μ, find the decomposition of their tensor product ''L''(λ) <math>\otimes </math> ''L''(μ) into irreducible representations.

* Suppose that <math>\mathfrak{g}_1</math> is the [[Levi decomposition|Levi component]] of a [[parabolic subalgebra]] of a semisimple Lie algebra <math>\mathfrak{g}</math>. For a given dominant highest weight '''&lambda;λ''', determine the [[branching rule]] for decomposing the restriction of ''L''('''&lambda;λ''') to <math>\mathfrak{g}_1</math>.<ref>Every complex semisimple Lie algebra <math>\mathfrak{g}</math> is the [[complexification]] of the Lie algebra of a compact connected simply connected semisimple Lie group. The subalgebra <math>\mathfrak{g}_1</math> corresponds to a maximal rank closed subgroup, i.e. one containing a maximal torus.</ref>

(Note that the first problem, of weight multiplicities, is the special case of the third in which the parabolic subalgebra is a Borel subalgebra. Moreover, the Levi branching problem can be embedded in the tensor product problem as a certain limiting case.)