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dykinin

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There are 7 letters in DYKININ ( D2I1K5N1Y4 )

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Dykinin might refer to
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
*
* +
* ...
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