ID | 60871 |
FullText URL | |
Author |
Seita, Kohei
Department of Mathematics, Graduate School of Natural Science and Technology, Okayama University
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Abstract | Let G be a finite group and let V and W be real G-modules. We call V and W dim-equivalent if for each subgroup H of G, the H-fixed point sets of V and W have the same dimension. We call V and W are Smith equivalent if there is a smooth G-action on a homotopy sphere Σ with exactly two G-fixed points, say a and b, such that the tangential G-representations at a and b of Σ are respectively isomorphic to V and W . Moreover, We call V and W are d-Smith equivalent if they are dim-equivalent and Smith equivalent. The differences of d-Smith equivalent real G-modules make up a subset, called the d-Smith set, of the real representation ring RO(G). We call V and W P(G)-matched if they are isomorphic whenever the actions are restricted to subgroups with prime power order of G. Let N be a normal subgroup. For a subset F of G, we say that a real G-module is F-free if the H-fixed point set of the G-module is trivial for all elements H of F. We study the d-Smith set by means of the submodule of RO(G) consisting of the differences of dim-equivalent, P(G)-matched, {N}-free real G-modules. In particular, we give a rank formula for the submodule in order to see how the d-Smith set is large.
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Keywords | Real G-module
Smith equivalence
representation ring
Oliver group
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Note | Mathematics Subject Classification. Primary 57S25, Secondary 20C15.
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Published Date | 2021-01
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Publication Title |
Mathematical Journal of Okayama University
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Volume | volume63
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Issue | issue1
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Publisher | Department of Mathematics, Faculty of Science, Okayama University
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Start Page | 153
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End Page | 165
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ISSN | 0030-1566
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NCID | AA00723502
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Content Type |
Journal Article
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language |
English
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Copyright Holders | Copyright©2021 by the Editorial Board of Mathematical Journal of Okayama University
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File Version | publisher
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Refereed |
True
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Submission Path | mjou/vol63/iss1/9
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