Relationship between fans and root data

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)?

• Isn't there a pretty big difference between these two in that the algebraic groups people study from root datum are affine varieties, whereas the toric varieties are usually projective varieties? May 5 '20 at 13:55
• Yes. I agree it’s a long shot. Also, Sam I did lumina with you back in the day. My older sister is Lauren Teixeira; she was in Trojan Women with you when you were Poseidon. Hi I haven’t talked to you in a decade 👋🏼 May 5 '20 at 14:12
• Oh my god, I recognized the last name but didn't think that connection was possible. Way off topic, but I think saw an interview your sister did with a Chinese stunt drinker, and it was incredible. May 5 '20 at 14:17
• Haha ya that guy is wild but his wife made him stop the stunt drinking because she was worried about his health so now he just makes motivational videos for his fans, he’s actually a totally sweetheart. And ya the one thing that was giving me hope about affine and projective things somehow having the same classification is the example of symmetric spaces where the spaces of compact and noncompact type are as different as possible but have the same classification by duality May 5 '20 at 15:04
• Title of the reference, mostly recoverable from the URL but just in case of link rot: Rex - Linear algebraic monoids. Oct 29 '20 at 3:06

(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.

Not an answer, but: you can construct a fan from a root system. Let $$R$$ be a root system in an Euclidean space, and let $$\Lambda_R$$ be the root lattice with dual lattice $$\Lambda_R^\vee$$. The fan $$\Sigma$$ in $$\Lambda_R^\vee$$ associated to $$R$$ consists of the Weyl chambers of $$R$$ and all their faces. For instance, if $$R=A_1$$, then the associated toric variety is $$\mathbf{P}^1$$. I don't know how to determine when a fan comes from a root system, but I'm guessing someone here does.

• These are called "permutohedral varieties," I believe. May 5 '20 at 16:58
• Well in Type A they are called permutohedral varieties, at least; in other types maybe they would be called $W$-permutohedral varieties. May 5 '20 at 16:59