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Geometric representations of finite groups on real toric spaces

최수영|조수진|Shizuo Kaji
대한수학회 (The Korean Mathematical Society)|2019.09|대한수학회지 (Journal of the KMS)|제56권 제5호|pp. 1265-1283|DOI 10.4134/JKMS.j180646

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키워드

representationreal toric varietyWeyl groupposet topologySpecht modulebuilding setnestohedron
주제분류수학

초록

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.

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.

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