### The Toric Variety Associated to the Weyl Chambers

#### Posted by John Baez

Happy New Year! I’m making a resolution to avoid starting work on new papers. I want to spend more time learning math, playing music, and having fun.

For a long time, whenever people said the phrase toric variety, I’d cover my ears and refuse to listen to what they had to say. Since I knew nothing of algebraic geometry, I thought ‘toric varieties’ were just one more of those specialized concepts — like Fano varieties and del Pezzo surfaces — that were devised solely for the purpose of demonstrating one’s superior erudition in this esoteric subject.

That’s changed. Now I love toric varieties, and I have a question about them.

But I won’t explain what they are, because then *everyone would know*.

Just kidding. I do hope to explain toric varieties someday. As usual, they’re a lot nicer and a lot easier to understand than anybody admits. Jim Dolan and I have been busy learning about them over the last month. But I’m not quite ready to give them away yet.

Well… I can’t resist just giving away just one one clue: *toric varieties are the varieties you can define using equations that don’t involve addition or subtraction — just multiplication!* So, they fit in very nicely with Deitmar’s approach to the field with one element, where he generalizes algebraic geometry from commutative rings to commutative monoids, by seeing what we can do without addition. Addition turns out to be vastly overrated. Doing algebraic geometry without addition is an incredibly simple, beautiful idea — so now I love toric varieties!

(Hmm, maybe this why Deitmar he wrote a paper on F_{1}-schemes and toric varieties! I hadn’t really noticed that before. The sneaky devil!)

Anyway: in the usual game, people like to construct a toric variety from something called a ‘fan’. (I’ll explain this someday, honest — but not today.) And, any simple Lie algebra gives an obvious fan, consisting of the Weyl chambers in its Cartan subalgebra. So, there’s a way to get a toric variety from a simple Lie algebra! This should be important and interesting.

But: **what are these toric varieties like?** Can you get them from simple Lie algebras some *other* way?

There seems to be a paper on this subject:

- Claudio Procesi, The toric variety associated to Weyl chambers, Mots, 153-161, Lang. Raison. Calc., Hermès, Paris, 1990.

But, I don’t know how to get ahold of it.

Math Reviews has this to say:

Given a root system $\Phi$ in a finite-dimensional real Euclidean space $V$, let $W$ be the Weyl group and $M$ the $\mathbb{Z}$-lattice in $V$ generated by $\Phi$. Then

$N = \{v\in V | (v,\alpha)\in \mathbb{Z} for all \alpha\in\Phi\}$

is the lattice in $V$ dual to $M$, where the parentheses denote the inner product in $V$. Each set $\Sigma$ of simple roots in $\Phi$ determines a Weyl chamber

$C_{\Sigma} = \{v\in V| (v,\alpha)\geq 0 for all \alpha\in\Sigma \}$

and the set $\Delta$ of the faces of the Weyl chambers is a fan for $N$. The set $\Psi$ of fundamental weights is exactly the set of primitive elements in $N$ spanning the one-dimensional cones in the fan $\Delta$.

The fan $\Delta$ for $N$ determines a (complex) toric variety $X$, which turns out to be smooth and projective. Since the Weyl group $W$ is an automorphism group of the fan $\Delta$, the graded cohomology ring $H^*(X,\mathbb{Q})$ of $X$ with rational coefficients has a natural $W$-module structure.

The author describes the character of the $W$-module in terms of other $W$-modules. The key ingredients of the proof are as follows: (1) description, due to Jurkiewicz and Danilov [cf. V. I. Danilov, Uspekhi Mat. Nauk

33(1978), no. 2(200), 85–134, 247; MR0495499 (80g:14001) (Theorem 10.8 and Remark 10.9)], of $H^*(X,\mathbb{Q})$ as a quotient of the Stanley-Reisner ring $\overline{A}$ of the simplicial sphere determined by the fan $\Delta$; (2) $\overline{A}$ is a free module over the symmetric algebra $B$ of the $\mathbb{Q}$-vector space $M\otimes_{\mathbb{Z}}\mathbb{Q}$; (3) the determination, due to A. M. Garsia and D. W. Stanton [Adv. Math.51(1984), no. 2, 107–201; MR0736732 (86f:20003)], of the character of the natural $W$-module structure on the Stanley-Reisner ring $\overline{A}$.For the root systems $A_n$, the author also gives a recursive formula with respect to $n$.

Hmm. It seems Procesi’s fan consists of the set of *faces* of the Weyl chambers, instead of the Weyl chambers themselves. Both fans sound interesting. But either way, this stuff doesn’t make me go *“Aha! Now I understand the meaning of this toric variety!”*

Now I wish I hadn’t put my hands over my ears every time someone told me anything about toric varieties. I’m sorry… I apologize. Help!

## Re: The Toric Variety Associated to the Weyl Chambers

Isn’t this toric variety the GIT quotient of the conjugation action of the adjoint form of G on the wonderful compactification of G?