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
∑ w , w ′ ∈ W ϵ ( w w ′ ) P ( w ( λ + ρ ) + w ′ ( μ + ρ ) − ( ν + 2 ρ ) ) {\displaystyle \sum _{w,w^{\prime }\in W}\epsilon (ww^{\prime })P(w(\lambda +\rho )+w^{\prime }(\mu +\rho )-(\nu +2\rho ))}
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 Bourbaki, Nicolas (2005) [1975], Lie groups and Lie algebras. Chapters 7–9, Elements of Mathematics (Berlin), Berlin, New York: Springer-Verlag, ISBN 978-3-540-68851-8, MR 2109105 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, MR 0126508
