Steinberg formula

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

$\sum _{w,w^{\prime }\in W}\epsilon (ww^{\prime })P(w(\lambda +\rho )+w^{\prime }(\mu +\rho )-(\nu +2\rho ))$

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