Geometric representations of finite groups on real toric spaces
J. Korean Math. Soc. 2019 Vol. 56, No. 5, 1265-1283
https://doi.org/10.4134/JKMS.j180646
Published online September 1, 2019
Soojin Cho, Suyoung Choi, Shizuo Kaji
Ajou University; Ajou University; Kyushu University
Abstract : We develop a framework to construct geometric representations of finite groups $G$ through the correspondence between real toric spaces $X^\R$ 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
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