TITLE: Is the quotient of a toric variety by a finite group still toric
QUESTION [14 upvotes]: I have asked this question on Math.StackExchange, but haven't got any reply.
Suppose $X$ is a toric variety with fan $\Sigma$, and the lattice of one-parameter subgroups of its torus is $N$. Suppose $\phi$ is a representation of finite group $G$ in $N \otimes_{\mathbb{Z}} \mathbb{R}$,
\begin{equation}
\phi:G \rightarrow \text{GL}( N \otimes_{\mathbb{Z}} \mathbb{R})
\end{equation}
which sends the lattice $N$ to $N$, and the cones of $\Sigma$ to cones of $\Sigma$, therefore each $\phi(g)$ defines a toric automorphism of $X$.
Question 1: Is the quotient variety $X/G$ (which exists from GIT) still a toric variety? Is the quotient morphism $X \rightarrow X/G$ toric?
Question 2: If so, how to construct the fan of $X/G$ and the toric morphism $X \rightarrow X/G$?
Any references on the two questions?

REPLY [36 votes]: No. If $X$ is the 47-dimensional torus and $G=\mathbb{Z}/47\mathbb{Z}$ acting by the permutation representation, then $X/G$ is not a rational variety (R. Swan, Inv. math. 7, 148-158 (1969)), therefore not toric.