Journal of the
Korean Mathematical Society JKMS

We develop a framework to construct geometric representations of finite groups $G$ through the correspondence between real toric spaces $X^\R$ and simplicial complexes with characteristic matrices. We give a combinatorial description of the $G$-module structure of the homology of $X^\R$. As applications, we make explicit computations of the Weyl group representations on the homology of real toric varieties associated to the Weyl chambers of type~$A$ and $B$, which show an interesting connection to the topology of posets. We also realize a certain kind of Foulkes representation geometrically as the homology of real toric varieties.

Keywords: real toric variety, Weyl group, representation, poset topology, Specht module, building set, nestohedron

MSC numbers: Primary 05E10, 55U10, 14M25, 20C30; Secondary 05E25

Supported by: This work was supported by the Ajou University research fund.
The second named author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science, ICT & Future Planning(NRF-2016R1D1A1A09917654).
The third named author was partially supported by KAKENHI, Grant-in-Aid for Scientific Research (C) 18K03304.