Steinberg formula

Short description: Formula in representation theory

In mathematical representation theory, Steinberg's formula, introduced by Steinberg (1961), describes the multiplicity of an irreducible representation of a semisimple complex Lie algebra in a tensor product of two irreducible representations. It is a consequence of the Weyl character formula, and for the Lie algebra sl₂ it is essentially the Clebsch–Gordan formula.

Steinberg's formula states that the multiplicity of the irreducible representation of highest weight ν in the tensor product of the irreducible representations with highest weights λ and μ is given by

[math]\displaystyle{ \sum_{w,w^\prime\in W} \epsilon(ww^\prime)P(w(\lambda+\rho)+w^\prime(\mu+\rho)-(\nu+2\rho)) }[/math]

where W is the Weyl group, ε is the determinant of an element of the Weyl group, ρ is the Weyl vector, and P is the Kostant partition function giving the number of ways of writing a vector as a sum of positive roots.

References

• Steinberg, Robert (1961), "A general Clebsch–Gordan theorem", Bulletin of the American Mathematical Society 67 (4): 406–407, doi:10.1090/S0002-9904-1961-10644-7, ISSN 0002-9904