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LieTypes :: weightDiagram

weightDiagram -- computes the weights in a Lie algebra module and their multiplicities

Synopsis

Description

This function implements Freudenthal's recursive algorithm; see Humphreys, Introduction to Lie Algebras and Representation Theory, Section 22.3. Let $V$ be the irreducible $\mathbf{g}$-module with highest weight $v$. This function returns a hash table whose keys are the weights appearing in $V$ and whose values are the multiplicities of these weights. The character of $V$ can be easily computed from this information (but characters of Lie algebra modules have not been implemented in this version of LieTypes).

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

o1 = g

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

o2 = V

o2 : g module
 
i3 : weightDiagram(V)

o3 = HashTable{{-1, -2} => 1}
               {-1, 1} => 2
               {-2, 0} => 1
               {-2, 3} => 1
               {-3, 2} => 1
               {0, -1} => 2
               {0, 2} => 1
               {1, -3} => 1
               {1, 0} => 2
               {2, -2} => 1
               {2, 1} => 1
               {3, -1} => 1

o3 : HashTable

See also

Ways to use weightDiagram :

For the programmer

The object weightDiagram is a method function.