On Fri, Jun 19, 2020 at 10:14 AM John Cremona <john.crem...@gmail.com> wrote:
>
> On Fri, 19 Jun 2020 at 09:39, Dima Pasechnik <dimp...@gmail.com> wrote:
> >
> >
> >
> > On Fri, 19 Jun 2020, 09:05 John Cremona, <john.crem...@gmail.com> wrote:
> >>
> >> I would like to vote for exponentiation for a right group action. This is
> >> standard mathematical notation (as has already been said), at least in
> >> several fields that I know of. Using the shift operators looks too much
> >> like a computer-sciency solution to me. Sage has always tried to keep its
> >> notation close to what a mathematician (including students) would expect
> >> -- for example (almost the only one), re-purposing ^ to mean
> >> exponentiation rather than the python default.
> >
> >
> > I don't know how to reconcile this with group theory use of exponentiation
> > for the conjugation action(s).
>
> I had not thought of that. But if you want to write h^g for the
> conjugate of h by g and you want this to be a right action, so that
> g^(h1*h2) = (g^h1)^h2, then that is fine provided that "the conjugate
> of h by g" means g^(-1)*h*g.
>
> It's really the same issue. If instead you think of "the conjugate of
> h by g" as meaning g*h*g^(-1) the you have a left action and so the
> notation h^g is *not* appropriate! I would write that instead using
> left exponentiation, i.e. ${}^gh = ghg^{-1}$ in tex notation.
My problem is that if a group acts on itself, conjugation actions (be
it h|->ghg^-1 or h|->g^-1 hg)
are vastly different from regular actions (left or right).
How are you going to use '^' for both?
>
> Are you now reconciled, Dima?
>
> John
>
> >
> >>
> >> All students learning permutations get confused about the order of
> >> multiplcation and just need to learn that there are two conventions and
> >> you *have to* know which is being used when you read something since
> >> otherwise half of what you read will not make sense.
> >>
> >> When I was a student I had two tutors who I handed work in to each week.
> >> One was (is) a group theorist
> >> (https://en.wikipedia.org/wiki/Peter_M._Neumann) for whom we had to write
> >> xf for the result of applying function f to object x, so fg meant "do f
> >> then g". The other was/is an analyst who only allowed f(x) and the other
> >> convention. I had to keep both happy (and get the right answers) so I
> >> leant fast.
> >>
> >> John
> >>
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