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

## Synopsis

• Usage:
weightDiagram(V)
• Inputs:
• Outputs:
• T, ,

## 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

## Ways to use weightDiagram :

• "weightDiagram(LieAlgebraModule)"
• "weightDiagram(String,ZZ,List)"

## For the programmer

The object weightDiagram is .