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Define $\eta\uparrow\lambda$ if $\eta=\lambda$ or $\eta=s_\alpha\cdot\lambda<\lambda$ for some $\alpha\in\Phi^+$. More generally, $\eta\uparrow\lambda$ if $\eta=\lambda$ or $\eta=s_{\alpha_1}s_{\alpha_2}\cdots s_{\alpha_r}\cdot\lambda\uparrow s_{\alpha_2}\cdots s_{\alpha_r}\cdot\lambda \uparrow \cdots \uparrow s_{\alpha_r}\lambda\uparrow\lambda$ for some $\alpha_1, \alpha_2,\cdots, \alpha_r\in\Phi^+$.

Let $\lambda\in\mathfrak{h}^*$ and $X(\lambda)$={$\eta$$\in\mathfrak{h}^*$$:\eta\uparrow\lambda\}$.

I would like to ask whether there is any geometry about $X(\lambda)$ or the convex hull of $X(\lambda)$.

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    $\begingroup$ Can one make this more concrete in type $A_n$? $\endgroup$ Aug 16, 2019 at 12:19
  • $\begingroup$ If $\lambda$ is dominant, the convex hull of $X(\lambda)$ is the convex hull of $W(\lambda)$, i.e., a so-called $W$-permutohedron. Is this the kind of thing you are interested in? $\endgroup$ Aug 16, 2019 at 14:06
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    $\begingroup$ Isn't this (that the up arrow relation is the same as (strong) Bruhat order) exactly what your previous question was about?: mathoverflow.net/questions/338226/… $\endgroup$ Aug 16, 2019 at 14:34
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    $\begingroup$ The dot action part is a distraction, imo: that just has to do with replacing $\lambda$ by $\lambda-\rho$. Can't you still show by the same argument that for antidominant, regular $\lambda$, $s_{\alpha}w\lambda \uparrow w\lambda$ if and only if $s_{\alpha}w < w$? $\endgroup$ Aug 17, 2019 at 2:55
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    $\begingroup$ I might be getting dominant and antidominant mixed up, yes. $\endgroup$ Aug 17, 2019 at 18:29

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