next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
LieTypes :: multiplicity(List,LieAlgebraModule)

multiplicity(List,LieAlgebraModule) -- compute the multiplicity of a weight in a Lie algebra module

Synopsis

Description

This function implements Freudenthal’s recursive algorithm; see Humphreys, Introduction to Lie Algebras and Representation Theory, Section 22.3. This function returns the multiplicity of the weight v in the irreducible Lie algebra module M. For Type A (that is, g = slk), these multiplicities are related to the Kostka numbers (though in this package, irreducible representations are indexed by the Dynkin labels of their highest weights, rather than by partitions).

The example below shows that the sl3 module with highest weight (2,1) contains the weight (-1,1) with multiplicity 2.

i1 : g=simpleLieAlgebra("A",2)

o1 = g

o1 : LieAlgebra
i2 : V=irreducibleLieAlgebraModule({2,1},g)

o2 = V

o2 : g module
i3 : multiplicity({-1,1},V)

o3 = 2

See also