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

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