Geodesic
Toric Surfaces with Reflection Symmetries
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Toric Topology, Group Actions, Geometry, and Combinatorics. Part 2, Tome 318 (2022), pp. 177-192

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Let $W$ be a reflection group in a plane and $P$ a rational polygon that is invariant under the $W$-action. The action of $W$ on $P$ induces a $W$-action on the toric variety $X_P$ associated with $P$. In this paper, we study the $W$-representation on the cohomology $H^*(X_P)$ and show that the invariant subring $H^*(X_P)^W$ is isomorphic to the cohomology ring of the toric variety associated with the fundamental region $P/W$. As an example, we provide an explicit description of the main result for the case of the toric variety associated with the fan of Weyl chambers of type $G_2$.
Keywords: toric variety, toric surface, reflection, singular cohomology. 
@article{TM_2022_318_a10,
     author = {Jongbaek Song},
     title = {Toric {Surfaces} with {Reflection} {Symmetries}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {177--192},
     publisher = {mathdoc},
     volume = {318},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2022_318_a10/}
}
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Jongbaek Song. Toric Surfaces with Reflection Symmetries. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Toric Topology, Group Actions, Geometry, and Combinatorics. Part 2, Tome 318 (2022), pp. 177-192. http://geodesic.mathdoc.fr/item/TM_2022_318_a10/