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.
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:[1][2] (Note that the first problem, of weight multiplicities, is the special case of the third in which the parabolic subalgebra is a Borel subalgebra.
Answers to these questions were first provided by Hermann Weyl and Richard Brauer as consequences of explicit character formulas,[4] followed by later combinatorial formulas of Hans Freudenthal, Robert Steinberg and Bertram Kostant; see Humphreys (1994).
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.
n:[5][6][7][8] Attempts at finding similar algorithms without overcounting for the other classical Lie algebras had only been partially successful.
[9] 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
The combinatorial data could be encoded in a coloured directed graph, with labels given by the simple roots.
Littelmann's main motivation[10] 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 Hong & Kang (2002, p. xv).
In the case of complex semisimple Lie algebras, there is a simplified self-contained account in Littelmann (1997) relying only on the properties of root systems; this approach is followed here.
Let P be the weight lattice in the dual of a Cartan subalgebra of the semisimple Lie algebra
Let π(t) be a path lying wholly within the positive Weyl chamber defined by the simple roots.
If α is a simple root and k = h(1), with h as above, then the corresponding reflection sα acts as follows: If π is a path lying wholly inside the positive Weyl chamber, the Littelmann graph
is defined to be the coloured, directed graph having as vertices the non-zero paths obtained by successively applying the operators fα to π.
τ lies entirely in the positive Weyl chamber and the concatenation π