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LieTypes :: tensorCoefficient

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



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 ⊗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=sl3 and λ=2 ω1 + ω2, μ= ω1 + 2 ω2, and ν= 2 ω1 + 2 ω2, the tensor product of sl3 modules Vλ ⊗Vμ contains two copies of Vν.

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

o1 = g

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

o2 = U

o2 : g module
i3 : V=irreducibleLieAlgebraModule({1,2},g)

o3 = V

o3 : g module
i4 : W=irreducibleLieAlgebraModule({2,2},g)

o4 = W

o4 : g module
i5 : tensorCoefficient(U,V,W)

o5 = 2

See also

Ways to use tensorCoefficient :