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

Synopsis

Function: multiplicity
Usage:

multiplicity(v,M)
Inputs:
- v, a list,
- M, an instance of the type LieAlgebraModule,
Optional inputs:
- BasisElementLimit => ..., default value infinity,
- DegreeLimit => ..., default value {},
- MinimalGenerators => ..., default value true,
- PairLimit => ..., default value infinity,
- Strategy => ..., default value null,
- Variable => ..., default value "w",
Outputs:
- k, an integer,

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, $g = sl_k$), 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 $sl_3$ 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

Ways to use this method: