casimirSpectrum -- computes the eigenvalues of the Casimir operator associated to a representation

Usage:

casimirOperator(V)
Inputs:
- V, an instance of the type LieAlgebraModule,
Outputs:
- M, a matrix,

Description

Let V be a LieAlgebraModule, and recall the definition of the Casimir operator from casimirOperator.

If $V$ corresponds to an irreducible $\mathfrak{g}$-module with highest weight $\lambda$, then $\operatorname{Cas} = c(\lambda) \operatorname{Id}$, where $c(\lambda)$ is the scalar computed by casimirScalar.

This function returns a nonredundant list of eigenvalues of $\operatorname{Cas}$ by computing the scalars $c(\lambda)$ for each irreducible summand in $V$, and then removing any duplicates.

i1 : sl3 = simpleLieAlgebra("A",2);

i2 : V=standardModule(sl3);

i3 : S3V=symmetricPower(3,V);

i4 : S4S3V=symmetricPower(4,S3V);

i5 : casimirSpectrum(S4S3V)

o5 = {0, 16, 24, 30, 36, 48, 60, 76, 120}

o5 : List

Ways to use casimirSpectrum:

casimirSpectrum(LieAlgebraModule)

For the programmer

The object casimirSpectrum is a method function.

The source of this document is in LieAlgebraRepresentations/documentation.m2:1873:0.