Geometric representations of finite groups on real toric spaces
J. Korean Math. Soc.
Published online July 17, 2019
Soojin Cho, Suyoung Choi, and Shizuo Kaji
Ajou University, Kyushu University
Abstract : We develop a framework to construct geometric representations of finite groups $G$
through the correspondence between real toric spaces $X^{\mathbb R}$ and simplicial complexes with characteristic matrices.
We give a combinatorial description of the $G$-module structure of the homology of $X^{\mathbb R}$.
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
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