On Mon, Jul 15, 2013 at 10:34 AM, Peter Bruin <pjbr...@gmail.com> wrote:
> Hi Marco and all,
>
>> I had Darij's problem as well, and many others probably did as well.
>> In a right action, I would prefer p(1) to give a warning. In a right
>> action, I would want some notation where p is on the right, preferably
>> 1^p (1 hat p).
>
>
> That would make sense (except that I don't really see why "^" is better than
> "*", see below). In principle one can even allow completely symmetric
> notation:
>
> - left action of g on x: g(x) or g^x; think of [left exponent g]x in
> two-dimensional notation
Trivial remark: I don't think anybody is suggesting that we use
exponentiation to denote a *left* action. Above, he wrote " In a
right action, I would want ...".
> - right action of g on x: (x)g or x^g
>
> Of course g^x and (x)g look a bit funny and maybe too confusing, but this is
> just because we are used to thinking that g^x means that x is in the
> exponent (as opposed to g, on the left), and we are not used to (x)g at all.
> I guess existing parsers could be enhanced to accept all these notations if
> somebody is crazy enough to want them. 8-)
>
>> The notation "*" has the wrong distributive laws in case of actions on
>> rings or groups. Of course this is irrelevant for permutations acting
>> on sets, but since Galois groups can be interpreted as permutation
>> groups too and they act on rings, the hat is much better.
>
>
> For both left and right actions, whether multiplicative ("*", similar binary
> symbols or the empty notation) or exponential notation ("^", left or right
> exponents) looks more natural depends on whether you are looking at the
> behaviour of the group action with respect to addition or with respect to
> multiplication. The following (and their equivalents for right actions)
> look OK:
>
> g*(x + y) = g*x + g*y
> [left exponent g](x*y) = [left exponent g]x * [left exponent g]y
> g^(x*y) = (g^x)*(g^y) (as long as you think of g as the exponent, not x and
> y)
>
> But the following look somewhat less appropriate:
> g*(x*y) = (g*x)*(g*y)
> [left exponent g](x + y) = [left exponent g]x + [left exponent g]y
> (especially strange for right actions)
> g^(x + y) = g^x + g^y
>
> Peter
>
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--
William Stein
Professor of Mathematics
University of Washington
http://wstein.org
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