Littelmann path model
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In the mathematical field of representation theory, the Littelmann path model is a combinatorial model for irreducible representations of semisimple Lie algebras. It provides combinatorial answers to the following major questions of representation theory:
- For a given dominant weight λ, find the character formula for the simple highest weight module L(λ).
- For two simple highest weight modules with dominant highest weights μ and ν, find the decomposition of their tensor product into simple summands.
- Suppose that l is a Levi subalgebra of a semisimple Lie algebra g. For a given dominant highest weight λ, determine the decomposition of L(λ) under the restriction from g to l.
Prior to Littlemann's work, the answers to the first two questions had been known for the special linear Lie algebra g = sln ('type A' case) and, to a lesser extent, for other classical Lie algebras. The theory of canonical bases, due to George Lusztig and Masaki Kashiwara, addresses the same questions from a constructive point of view. Littlemann's contribution was to give a unified combinatorial model for representations that applied for all symmetrizable Kac-Moody algebras and provided explicit combinatorial algorithms for the multiplicities of weights, Littlewood-Richardson coefficients, and the restriction multiplicities. This was accomplished by introducing a certain vector space spanned by (piecewise-linear) 'paths' and a family of linear operators on this vector space indexed by the roots of g, called the root operators.
[edit] Definition
A Littelmann path is a piecewise-linear function π from [0,1] to h*, where h is the Cartan subalgebra of a Lie algebra g, such that π(0) = 0 and π(1) is a weight.
[edit] See also
[edit] References
- Littelmann, Peter, A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras. Invent. Math. 116 (1994), no. 1-3, 329--346 MR1253196
- Littelmann, Peter, Paths and root operators in representation theory. Ann. of Math. (2) 142 (1995), no. 3, 499--525 MR1356780
