title: Steinberg formula
text: # 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 $sl_2$ 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.
  
  