Abstract: We study toric varieties associated with $W$-permutohedra and their quotients by parabolic subgroups. For a Weyl group $W$ and a $W$-permutohedron $P$, the quotient $P/W_K$ by a parabolic subgroup $W_K$ can be identified with a polytope inside $P$, giving rise to toric varieties $X_P$ and $X_{P/W_K}$.

We construct an explicit algebra isomorphism between the rational cohomology ring of $X_{P/W_K}$ and the invariant of the rational cohomology ring of $X_P$$$H^*(X_{P/W_K};\mathbb{Q})\cong H^*(X_P;\mathbb{Q})^{W_K},$$ and generalize this result to non-degenerate $W$-symmetric polytopes, to intermediate lattices, and to finite Coxeter groups via a polytopal algebra model.

From a topological perspective, we prove that $X_P/W_K$ is homotopy equivalent to $X_{P/W_K}$, and extend this result to non-degenerate $W$-symmetric polytopes. In joint work with Crowley and Simpson, we further establish a variety isomorphism between $X_P/W_K$ and $X_{P/W_K}$, and generalize this isomorphism to $W$-symmetric polytopes.

These results give affirmative answers to questions of Horiguchi--Masuda--Shareshian--Song concerning equivalences between $X_P/W_K$ and $X_{P/W_K}$.

Finally, we study the real locus $X_P^{\mathbb{R}}$ of the toric variety $X_P$. We show that the quotient $X_P^{\mathbb{R}}/W$ is contractible, and describe the homotopy type of $X_P^{\R}/W_K$ in low-dimensional cases.