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.
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:
Answers to these questions were first provided by Hermann Weyl and Richard Brauer as consequences of explicit character formulas, followed by later combinatorial formulas of Hans Freudenthal, Robert Steinberg and Bertram Kostant; see . An unsatisfactory feature of these formulas is that they involved alternating sums for quantities that were known a priori to be non-negative. Littelmann's method expresses these multiplicities as sums of non-negative integers "without overcounting". His work generalizes classical results based on Young tableaux for the general linear Lie algebra formula_5"n" or the special linear Lie algebra formula_6"n":
Attempts at finding similar algorithms without overcounting for the other classical Lie algebras had only been partially successful.
Littelmann's contribution was to give a unified combinatorial model that applied to all symmetrizable Kac–Moody algebras and provided explicit subtraction-free combinatorial formulas for weight multiplicities, tensor product rules and branching rules. He accomplished this by introducing the vector space "V" over Q generated by the weight lattice of a Cartan subalgebra; on the vector space of piecewise-linear paths in "V" connecting the origin to a weight, he defined a pair of "root operators" for each simple root of formula_3.
The combinatorial data could be encoded in a coloured directed graph, with labels given by the simple roots.
Littelmann's main motivation was to reconcile two different aspects of representation theory:
Although differently defined, the crystal basis, its root operators and crystal graph were later shown to be equivalent to Littelmann's path model and graph; see . In the case of complex semisimple Lie algebras, there is a simplified self-contained account in relying only on the properties of root systems; this approach is followed here.
Definitions.
Let "P" be the weight lattice in the dual of a Cartan subalgebra of the semisimple Lie algebra formula_3.
A Littelmann path is a piecewise-linear mapping
such that π(0) = 0 and π(1) is a weight.
Let ("H" α) be the basis of formula_15 consisting of "coroot" vectors, dual to basis of formula_15* formed by simple roots (α). For fixed α and a path π, the function formula_17 has a minimum value "M".
Define non-decreasing self-mappings "l" and "r" of [0,1] formula_18 Q by
Thus "l"("t") = 0 until the last time that "h"("s") = "M" and "r"("t") = 1 after the first time that "h"("s") = "M".
Define new paths πl and πr by
The root operators "e"α and "f"α are defined on a basis vector [π] by
The key feature here is that the paths form a basis for the root operators like that of a monomial representation: when a root operator is applied to the basis element for a path, the result is either 0 or the basis element for another path.
Properties.
Let formula_23 be the algebra generated by the root operators. Let π("t") be a path lying wholly within the positive Weyl chamber defined by the simple roots. Using results on the path model of C. S. Seshadri and Lakshmibai, Littelmann showed that
There is also an action of the Weyl group on paths [π]. If α is a simple root and "k" = "h"(1), with "h" as above, then the corresponding reflection "s"α acts as follows:
If π is a path lying wholly inside the positive Weyl chamber, the Littelmann graph formula_25 is defined to be the coloured, directed graph having as vertices the non-zero paths obtained by successively applying the operators "f"α to π. There is a directed arrow from one path to another labelled by the simple root α, if the target path is obtained from the source path by applying "f"α.
The Littelmann graph therefore only depends on λ. Kashiwara and Joseph proved that it coincides with the "crystal graph" defined by Kashiwara in the theory of crystal bases.
Applications.
Character formula.
If π(1) = λ, the multiplicity of the weight μ in "L"(λ) is the number of vertices σ in the Littelmann graph formula_26 with σ(1) = μ.
Generalized Littlewood–Richardson rule.
Let π and σ be paths in the positive Weyl chamber with π(1) = λ and σ(1) = μ. Then
where τ ranges over paths in formula_28 such that π formula_29 τ lies entirely in the positive Weyl chamber and
the "concatenation" π formula_29 τ (t) is defined as π(2"t") for "t" ≤ 1/2 and π(1) + τ( 2"t" – 1) for "t" ≥ 1/2.
Branching rule.
If formula_2 is the Levi component of a parabolic subalgebra of formula_3 with weight lattice "P"1 formula_33 "P" then
where the sum ranges over all paths σ in formula_25 which lie wholly in the positive Weyl chamber for formula_2.