Ralf Hemmecke <[EMAIL PROTECTED]> writes:
> I don't know why power sums are important for plethysm. I understand that
> they are somehow a motivation for doing all this, but otherwise even in the
> definition of Mishna I cannot see that p_n cannot be thought of as an
> ordinary *indeterminate*.
Well, they *can* be seen as an ordinary indeterminate: the power sums are
algebraically independent...
> 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]
>
> In which way is K[[p]] different from K[[x_1,x_2,x_3, ...]]? It is not.
quite right. As I said before, it is only a different point of view.
> And p_n[p_m] = p_{nm} is just another way of saying the index "m-stretching"
> for variables x_n.
yes.
> Seemingly, plethysm is commutative by definition, but my eyes don't see a
> definition of a plethysm of two arbitrary elements of K[[p]].
No, if I'm not completely mistaken, the plethysm is not commutative. Only for
power sums.
> 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.
Martin
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