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This is a rather technical question, it arose in connection of some calculations that I need to have better grasp of the question Formal group law over $\mathbb{F}_p$ and my own older one What is known about the sum x^{n^2}/n?.

Let me formulate it for one specific particular case I need, it will be clear what is the general situation I'm interested in.

Let $w_n$ be the Witt symmetric functions, defined usually by $\sum_ix_i^n=\sum_{d|n}dw_d(x_i)^{\frac nd}$. I need to express $w_n(w_1,w_2^2,w_3^3,...,w_k^k,...)^n$ in some of the many bases for symmetric functions, to have maximally explicit and manageable formula. Actually I also need to further iterate this, i. e. obtaining some symmetric function $f$ as a result, further plug into it $w_k^k$ in place of $x_k$. The basis is not that important although I somehow feel that finding an expression in the $w$ basis again would be better.

A straightforward approach is just to express each $w_n$ in terms of $x_i$, simply plug the $w_k^k$ in place of $x_k$ everywhere and recalculate, but I was not able to derive completely explicit formula along these lines.

So the question is - are there some methods to compute expressions like that above which avoid passing through variables? In general, how to reduce substitutions like above to computationally familiar constructions like plethysms, etc.? Is there a way to pick an optimal basis in which an expression like above would be most manageable? Specifically for the Witt symmetric functions above - is there any other basis in which their expansions can be explicitly written out?

Concerning the last one I am aware of papers by Reutenauer and by Scharf and Thibon where properties of the basis $r_*$ dual to $w_*$ are investigated and expressions of $r_*$ in terms of Schur functions are given, but I don't know how to use this.

PS

After posting, I found in the "Related" column a related question Reference or a short argument that a certain subset generates the ring of p-typical symmetric functions under plethysm which is unanswered so far - the generators $w_p^{\circ i}$ would be probably useful for me, except I don't know relations between them. I don't think they are algebraically independent, are they?

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    $\begingroup$ I don't know the answer to your main question. (It actually seems a bit bizarre to me. With plethysm, one substitutes the monomials of one symmetric function into the x-variables of the other. But that's not what you're doing.) Anyway, the $w_p^{\circ i}$ are algebraically independent as $p$ and $i$ vary. They don't generate the full ring of symmetric functions. (With $\mathbb{Q}$-coefficients, they only generated the subring generated by the power-sum symmetric functions of prime-power index.) $\endgroup$
    – JBorger
    Feb 15, 2016 at 10:50
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    $\begingroup$ But you can use them to make algebraically independent generators. The $\circ$-words $w_{p_1}\circ\cdots\circ w_{p_n}$ with $p_1\leq \cdots\leq p_n$ form an independent generating set, if I remember correctly. Here $\leq$ denotes any ordering of the prime numbers, such as the usual one. $\endgroup$
    – JBorger
    Feb 15, 2016 at 10:51
  • $\begingroup$ @JBorger Very interesting! It would be great if you could give me a reference for that. $\endgroup$ Feb 15, 2016 at 14:10

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