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I am interested in studying toric varieties associated to the fan of Weyl chambers. General information would be best but I am also interested in the specific case of the Weyl chamber of $\mathfrak{sl}(3)$!

One way to define this toric variety is: the closure of the torus in the wonderful compactification of $G$ (for example, $G=PSL(3)$). Another way is: the closure of the generic $T$-orbit in the flag variety $G/B$.

I am most interested in: 0) geometric description for lower-rank varieties, ex. attached to $\mathfrak{sl}(3)$

  1. explicit description of Picard group of these toric varieties (specifically in terms of $T$-invariant divisors, e.g. rays in the fan)
  2. some formula of Euler characteristic of line bundles on these varieties (again in terms of coefficients of $T$-invariant divisors)

One caveat regarding (1): I am aware of the short exact sequence describing the Picard group as the free lattice on the $T$-invariant divisors modulo the image of the functions under $div$ map, but a more concrete description would be very nice!

Thank you very much!

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    $\begingroup$ If $G$ is of adjoint type, then that toric variety also equals the geometric quotient of the conjugation action of $G$ on its wonderful compactification. Up to tensoring with $\mathbb{Q}$, the Picard group of that toric variety equals the Picard group of the wonderful compactification. This group is the free Abelian group on the irreducible components of the boundary divisor. These are indexed by simple positive roots. $\endgroup$
    – Jason Starr
    Commented Dec 27, 2022 at 12:41

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