On 02/16/2007 08:45 AM, Martin Rubey wrote:
> Ralf Hemmecke <[EMAIL PROTECTED]> writes:
>
>
>> Page 29 says:
>>
>> Plethysm is a way to compose symmetric functions. It can be defined on the
>> power sum ring basis and extended to all of K[[p]]. It is defined by
>> p_n[p_m] = p_{nm} and extended by
>>
>> (fg)[h] = f[h] g[h] (*)
>> (f + g)[h] = f[h] + g[h] (+)
>> p_n[g] = g[p_n]
>
>>>>> I can only see a definition f[g] where at least one (either f or g) is a
>>>> polynomial. Don't you agree?
>>> No. Expand f and g in terms of the power sum symmetric functions and apply
>>> the definitions above.
>> Sorry, but I don't agree. And you probably see it if you Assume that we would
>> not have K[[p]] but K[p]. The definition only says something about *finite*
>> sums.
>
> Hmm, I don't have the time to go through this in detail, but maybe the
> definition above is not entirely correct. Take the one from Stanley, EC2,
> A2.6,
> he defines f[p_n] for an arbitrary symmetric function f, and then extends as
> above...
As I read it Stanley defines plethysm for symmetric functions (which are
in fact formal power series in infinitely many variables where any
permutation of variables gives the same function). This is, of course
only a subset of K[[x1,x2,x3,...]]. Or should I read that \Lambda is
equal to K[[x1,x2,x3,...]]?
Anyway, he does the same as Mishna. Seemingly, he also has no problem in
using (A2.161) and (A2.162) (which are basically (+) and (*) from above)
to step from plethysm for symmetric functions to plethysm for *infinite*
sums of the basic symmetric functions.
I must admit that I am a bit hesitant to accept that _arbitrary finite_
is the same as _infinite_. Otherwise there would not be a distiction
between K[x] and K[[x]].
But perhaps simply the definition is not rigorous enough or I am to
blind to see that everything is fine.
Ralf
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