On Fri, 19 Jun 2020 at 10:24, Dima Pasechnik <dimp...@gmail.com> wrote:
>
> 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?
I was not: in fact I said "I would like to vote for exponentiation for
a right group action". Only for right actions! For left actions
we can use * (with the group element which is acting on the left of
the *), though that could be confusing if the group was acting on
itself on the left not by the left regular action (e.g. by
conjugation). Let's face it, a group can have many different left or
right actions on the same set so whatever symbol we choose, somewhere
we'll have to specify which action we mean. In a mathematical text
you just give the definition and that's it. Here too then the action
intended must be defined somewhere, deducible from the types/parents
of the objects concerned. In the context of the original post we have
a finite set {1,2,...,n} and the symmetric group acting on it. And
also some vector space and matrices acting: either on the left, on
column vectors, or on the right, on row vectors. For these, Sage uses
* for both and (as you know of course) does not distinguish between
row and column vectors except by context:
sage: v2 = vector([1,2])
sage: v2
(1, 2)
sage: v3 = vector([1,2,3])
sage: v3
(1, 2, 3)
sage: m23 = matrix([[11,12,13],[21,22,23]])
sage: m23
[11 12 13]
[21 22 23]
sage: v2*m23
(53, 56, 59)
sage: m23*v3
(74, 134)
(while of course v3*m23 and m23*v2 raise errors). This is all good.
If we were to allow ^ for right action of a matrix on a (row) vector
that would probably be OK but we certainly should not take away the
ability to use * as well for the same thing (as in m23*v3) or many
loud objections would be raised.
By the way, I think it may be true that the preparser changes every ^
into ** so if we did introduce ^ for a right group action for use
from the command line it would have to be implemented as **; for .py
files I hope we could actually implement ^ to be used as itself.
>
>
> >
> > 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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