Macaulay2 » Documentation
Packages » LieAlgebraRepresentations :: casimirSpectrum
next | previous | forward | backward | up | index | toc

# 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 /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/LieAlgebraRepresentations/documentation.m2:1873:0.