Weyl Character Formula - Freudenthal's Formula

Freudenthal's Formula

Hans Freudenthal's formula is a recursive formula for the weight multiplicities that is equivalent to the Weyl character formula, but is sometimes easier to use for calculations as there can be far fewer terms to sum. It states

((\Lambda+\rho)^2 - (\lambda+\rho)^2)\dim V_\lambda = 2 \sum_{\alpha>0}\sum_{j\ge 1} (\lambda+j\alpha, \alpha)\dim V_{\lambda+j\alpha}

where

  • Λ is a highest weight,
  • λ is some other weight,
  • dim Vλ is the multiplicity of the weight λ
  • ρ is the Weyl vector
  • The first sum is over all positive roots α.

Read more about this topic:  Weyl Character Formula

Famous quotes containing the word formula:

    “Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective positions of the beings which compose it, if moreover this intelligence were vast enough to submit these data to analysis, it would embrace in the same formula both the movements of the largest bodies in the universe and those of the lightest atom; to it nothing would be uncertain, and the future as the past would be present to its eyes.”
    —Pierre Simon De Laplace (1749–1827)