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On the standard $n$-simplex $\Delta[n]$ in $\mathbb{R}^{n+1}$ there is the usual action of the symmetric group $S_n$, by permuting the vertices. Any element $g\in S_n$ then acts on $\Delta[n]$ linearly, by a permutation matrix.

By definition of group action, a group element $g$ acts on the set $\Delta[n]$ by a bijection. Yet, without further assumptions, it is not clear that the action should be linear.

Question: Are there simple examples of a finite group action on $\Delta[n]$, which is not linear? Or linear, but not by permutations?

Shlomi A
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  • I'm not sure what you mean. There are many homeomorphisms of the unit interval, say. This group of homeomorphisms naturally acts on tbe (geometric realization of a) $1$-simplex. – HallaSurvivor Feb 16 '21 at 17:44
  • @HallaSurvivor, I'm interested in relatively-simple examples, of preferably finite group actions. From your comment, one could pick any homeomorphism of (say) the unit interval, its inverse, and define thus an $\mathbb{Z}_2$-action on it. – Shlomi A Feb 16 '21 at 17:55
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    You need to be a little bit careful. Choosing one element $f$ (and its inverse) doesn't necessarily give you a $\mathbb Z/2$ action. It gives you a $\mathbb Z/n$ action where $n$ is the order of $f$. If $f$ has infinite order (as I expect most of these do, though I haven't checked) then this gives you a $\mathbb Z$ action. – HallaSurvivor Feb 16 '21 at 17:58
  • That's still discrete, though, so maybe suffices for your purposes? If you want an explicit example, identify the unit interval with $[-\infty, \infty]$ and look at the $x \mapsto x+1$ homeomorphism. This is a nonlinear $\mathbb Z$ action. – HallaSurvivor Feb 16 '21 at 18:00
  • How about a finite group action? – Shlomi A Feb 16 '21 at 18:01
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    Again, after identifying the interval with $[-\infty, \infty]$, reflect around, any nonzero point. In $[0,1]$ this would look like the function sending $3/4$ to itself (say), swapping $0$ and $1$, then separately extending linearly on both sides. – HallaSurvivor Feb 16 '21 at 18:26
  • In particular, this would be of order 2. Thanks! – Shlomi A Feb 16 '21 at 19:33

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