Dynkin index

In mathematics, the Dynkin index $$I({\lambda})$$ of finite-dimensional highest-weight representations of a compact simple Lie algebra $$\mathfrak g$$ relates their trace forms via

$$ \frac{\text{Tr}_{V_\lambda}}{\text{Tr}_{V_\mu}}= \frac{I(\lambda)}{I(\mu)}.$$

In the particular case where $$\lambda$$ is the highest root, so that $$V_\lambda$$ is the adjoint representation, the Dynkin index $$I(\lambda)$$ is equal to the dual Coxeter number.

The notation $$\text{Tr}_V$$ is the trace form on the representation $$\rho: \mathfrak{g} \rightarrow \text{End}(V)$$. By Schur's lemma, since the trace forms are all invariant forms, they are related by constants, so the index is well-defined.

Since the trace forms are bilinear forms, we can take traces to obtain


 * $$I(\lambda)=\frac{\dim V_\lambda}{2\dim\mathfrak g}(\lambda, \lambda +2\rho)$$

where the Weyl vector


 * $$\rho=\frac{1}{2}\sum_{\alpha\in \Delta^+} \alpha$$

is equal to half of the sum of all the positive roots of $$\mathfrak g$$. The expression $$(\lambda, \lambda +2\rho)$$ is the value of the quadratic Casimir in the representation $$V_\lambda$$.