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dynkinindex
dynkin index
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There are 11 letters in DYNKININDEX ( D2E1I1K5N1X8Y4 )
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Definitions of dynkin index in various dictionaries:
DYNKIN INDEX - In mathematics, the Dynkin index x λ {\displaystyle x...
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In mathematics, the Dynkin index * * * * * x * * λ * * * * * {\displaystyle x_{\lambda }} * of a representation with highest weight * * * * * | * * λ * * | * * * * {\displaystyle |\lambda |} * of a compact simple Lie algebra * * * * * * g * * * * * {\displaystyle {\mathfrak {g}}} * that has a highest weight * * * * λ * * * {\displaystyle \lambda } * is defined by * * * * * * * t * r * * * ( * * t * * a * * * * t * * b * * * ) * = * 2 * * x * * λ * * * * g * * a * b * * * * * {\displaystyle {\rm {tr}}(t_{a}t_{b})=2x_{\lambda }g_{ab}} * evaluated in the representation * * * * * | * * λ * * | * * * * {\displaystyle |\lambda |} * . Here * * * * * t * * a * * * * * {\displaystyle t_{a}} * are the matrices representing the generators, and * * * * * * g * * a * b * * * * * {\displaystyle g_{ab}} * is given by * * * * * * * t * r * * * ( * * t * * a * * * * t * * b * * * ) * = * 2 * * g * * a * b * * * * * {\displaystyle {\rm {tr}}(t_{a}t_{b})=2g_{ab}} * evaluated in the defining representation. * By taking traces, we find that * * * * * * x * * λ * * * = * * * * dim * * * | * * λ * * | * * * * 2 * dim * * * * g * * * * * * ( * λ * , * λ * + * 2 * ρ * ) * * * {\displaystyle x_{\lambda }={\frac {\dim |\lambda |}{2\dim {\mathfrak {g}}}}(\lambda ,\lambda +2\rho )} * where the Weyl vector * * * * * ρ * = * * * 1 * 2 * * * * ∑ * * α * ∈ * * Δ * * + * ... |