0

Let $\alpha_i, i \in I$ be simple roots in a root system and $\omega_i, i \in I$ the fundamental weights. The fundamental roots are dual to the fundamental coweights and the fundamental weights are dual to the fundamental coroots: $\alpha_i(\omega_j^{\vee}) = \delta_{ij}$, $\omega_i(\alpha_j^{\vee})=\delta_{ij}$. What is $\alpha^{\vee}$ for any root $\alpha$ in the root system? For example, in type $A_2$, what is $(\alpha_1 + \alpha_2)^{\vee}$? Thank you very much.

LJR
  • 14,520
  • Usually $\alpha^\vee$ denotes $2\frac{\alpha}{(\alpha,\alpha)}$, where $(\bullet, \bullet)$ denotes the inner product of your root system. The set of coroots also generates a root system. – juan diego rojas Nov 06 '19 at 14:11
  • @JuanDiegoRojas, thank you very much. I would like to think $\alpha^{\vee}$ as an element in the space generated by $\alpha_i^{\vee}$ and express $\alpha^{\vee}$ as a linear combination of $\alpha_i^{\vee}$. – LJR Nov 06 '19 at 14:14
  • See https://math.stackexchange.com/questions/724491/how-to-prove-that-b-vee-is-a-base-for-coroots – juan diego rojas Nov 06 '19 at 14:18

0 Answers0