A (split) reductive linear algebraic group is equivalently described by combinatorial information called a root datum.


A toric variety is described by combinatorial information called a fan. 


Both correspondences use the character lattice. 


The reference:


<http://u.cs.biu.ac.il/~margolis/Linear%20Algebraic%20Monoids/Renner-%20Lin.%20Alg.%20Monoids.pdf>


says that spherical varieties are a nice class of objects that include 
all my favorite spaces (e.g. symmetric spaces, toric varieties) . And, moreover, that a spherical variety is equivalent to combinatorial information called a colored fan. Is there any way of recovering a root datum from a colored fan? Or is a reductive group actually given as part of the data of a colored fan?


Are fans/ Toric varieties and root data/ reductive groups both special cases of a larger pattern (for example, colored fans/ spherical varieties)?



Relationship between fans and root data
(1) A (connected) reductive group $G$ over an algebraically closed field $k$ is described by a combinatorial object called the *based root datum* ${\rm BRD}(G)$. 


(2) A *spherical homogeneous space* $Y=G/H$ is a homogeneous space on which a Borel subgroup $B$ of $G$ acts with an open Zariski-dense orbit. It is described (uniquely at least in characteristic 0) by its *homogeneous combinatorial invariants*. These combinatorial invariants constitute an additional structure on ${\rm BRD}(G)$. 


(3) A *spherical embedding* $G/H\hookrightarrow Y^e$ is a normal $G$-variety $Y^e$ containing a spherical homogeneous space $G/H$ as an open dense $G$-orbit. It is described by its *colored fan*, which is an additional structure on the homogeneous combinatorial invariants.


By *spherical varieties* one means spherical homogeneous spaces and spherical embeddings.


Therefore, I think that the based root datum of $G$ should be regarded as a part of data describing the $G$-variety $Y^e$.


In the case when $G=T$ is a torus, we take $H=1$, and then the spherical embeddings of $G/H=T$ are the same as the toric varieties for $T$, and the corresponding colored fans are just fans. 


Reference: [Nicolas Perrin, On the geometry of spherical varieties](https://arxiv.org/abs/1211.1277).


