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

# tensorCoefficient -- computes the multiplicity of W in U tensor V

- Usage:
    `tensorCoefficient(U,V,W)`
- Inputs:
  - U, an instance of the type LieAlgebraModule,
  - V, an instance of the type LieAlgebraModule,
  - W, an instance of the type LieAlgebraModule,
- Outputs:
  - k, an integer,

## Description

This function implements the Racah-Speiser algorithm; see Di Francesco, Mathieu, and Senechal, Conformal Field Theory, Springer Graduate Texts in Theoretical Physics, Section 13.5.2.

Given three irreducible Lie algebra modules $U$, $V$, and $W$, the function returns the multiplicity of $W$ in $U \otimes V$. In Type A, these are related to the Littlewood-Richardson coefficients (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 for $g=sl_3$ and $\lambda=2 \omega_1 + \omega_2$, $\mu= \omega_1 + 2 \omega_2$, and $\nu= 2 \omega_1 + 2 \omega_2$, the tensor product of $sl_3$ modules $V_{\lambda} \otimes V_{\mu}$ contains two copies of $V_{\nu}$.

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

    o1 = g

    o1 : simple LieAlgebra

    i2 : U=irreducibleLieAlgebraModule({2,1},g)

    o2 = U

    o2 : irreducible LieAlgebraModule over g

    i3 : V=irreducibleLieAlgebraModule({1,2},g)

    o3 = V

    o3 : irreducible LieAlgebraModule over g

    i4 : W=irreducibleLieAlgebraModule({2,2},g)

    o4 = W

    o4 : irreducible LieAlgebraModule over g

    i5 : tensorCoefficient(U,V,W)

    o5 = 2

## See also

- LieAlgebraModule ** LieAlgebraModule -- tensor product of LieAlgebraModules

## Ways to use tensorCoefficient:

- `tensorCoefficient(LieAlgebraModule,LieAlgebraModule,LieAlgebraModule)`

## For the programmer

The object tensorCoefficient is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/LieAlgebraRepresentations/documentation.m2:893:0.