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Let $G$ be a reductive group over a field $k$ of characteristic zero with maximal split torus $T$ of rank $n$ and a Borel $B \supset T$ defining a set of simple roots $\Delta$. By $X_*(T)$ we denote the cocharacter group.

I read that there is a dominance order on $X_*(T)$ with respect to $B$ but couldn't find any formal definition. As the dominance order is mentioned with respect to $B$, I guess that the simple roots are involved too.

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  • $\begingroup$ So it seems that then dominance order "$\geq$" on $X_*(T)$ is defined the following way. Let $\mu, \lambda \in X_*(T)$, then $\mu \geq \lambda$ iff $\mu-\lambda=\sum_{\alpha \in \Delta} n_{\alpha} \alpha^\vee{}$ for $n_\alpha \geq 0$. $\endgroup$
    – KKD
    Aug 31, 2021 at 10:54
  • $\begingroup$ That's right, though usually this is just called "the partial order on the weights" or something. The name you're seeing probably comes from the specific case of $G = \operatorname{GL}_n$, where the dominant integral weights are in bijection with partitions, and the partial order on weights agrees with the dominance order on partitions. $\endgroup$
    – Joppy
    Aug 31, 2021 at 15:19

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