Let $G$ be a finite group. Let $V_1, V_2$ be two finite-dimensional real representations. Suppose $f: V_1 \to V_2$ is a $G$-equivariant homeomorphism. Can one conclude that $V_1$ and $V_2$ are isomorphic representations?
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2$\begingroup$ Maybe you assume the group is finite? Otherwise take $\mathbf{Z}$ acting on $\mathbf{R}$ by powers of $x\mapsto 2x$, resp. $x\mapsto 3x$. $\endgroup$– YCorApr 5, 2022 at 21:48
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2$\begingroup$ This seems false for cyclic groups $\mathbb{Z}/n\mathbb{Z}$ acting on $\mathbb{C}$. There are conjugate characters. Their action is "the same up to conjugation," and so they are equivariantly homeomorphic, no? $\endgroup$– Geva YashfeApr 5, 2022 at 22:01
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1$\begingroup$ @GevaYashfe Are they isomorphic as real representations? $\endgroup$– UVIRApr 5, 2022 at 22:05
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1$\begingroup$ Sorry and thanks, I misread the question. $\endgroup$– Geva YashfeApr 5, 2022 at 22:07
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1$\begingroup$ @LSpice $x \mapsto \operatorname{sign}(x) |x|^{ \log 3/\log 2}$. $\endgroup$– Will SawinApr 5, 2022 at 22:10
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1 Answer
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This is a famous problem, originating in work of de Rham, and the answer turns out to be No. The lowest-dimensional examples of non-linear similarity, as it is called, are in dimension 6, and examples only exist if the group has order divisible by (but not equal to) 4. This article contains a summary of the subject:
Sylvain Cappell, Julius Shaneson, Mark Steinberger, Shmuel Weinberger, and James West. The classification of nonlinear similarities over ${\text{Z}}_{2^r}$, Bulletin of the American Mathematical Society 22 (1990).
Mark Steinberger wrote another short summary of the subject, available here: http://math.albany.edu/topics/steinberger/msteinbergerrsch.pdf.
answered Apr 6, 2022 at 0:41
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1$\begingroup$ Are you by chance able to provide an explicit group (e.g., cyclic of order 8) and and explicit pair of non-equivalent representations that are topologically conjugate? $\endgroup$– YCorApr 6, 2022 at 18:18
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1$\begingroup$ In principle I think one can work out an explicit example, e.g. for the cyclic group of order 8, from the information in Cappell and Shaneson's announcement projecteuclid.org/journals/… $\endgroup$ Apr 6, 2022 at 18:49
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1$\begingroup$ I hope someone will work out some such example and post it... $\endgroup$ Apr 6, 2022 at 18:50
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6$\begingroup$ Oh, it's explicit in the original Cappell-Shaneson paper. They provide many explicit examples, one of which (p318 pick $q=2$, $j=k=1$), for the cyclic group of order 8, is as follows: define $r_\theta$ as the rotation matrix of angle $2\pi\theta/8$ is given by the two 9-dimensional real representations given by powers of block-diagonal matrices $(r_1,r_1,r_1,r_1,-1)$ and $(r_5,r_5,r_5,r_5,-1)$. $\endgroup$– YCorApr 6, 2022 at 19:11
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