Details about plethysm I'm currently working on plethysm, i.e. the character of the composition $S^\lambda(S^\mu(V))$ of the Schur functors $S^\lambda$ and $S^\mu$ on a complex vector space $V$.  We note this character $s_\lambda[s_\mu]$. One must know that the character of $S^\mu(V)$ is the Schur function $s_\mu$.
We want to find the irreductible representation of $S^\lambda(S^\mu(V))$, or equivalently, to write $s_\lambda[s_\mu]$ in a sum of Schur function. There is very little that is known about this, and this operation is rarely mentionned in the litterature I found.
I have worked with plethysm mostly in the context of a "math research traineeship", so I skipped some details of all this theory to emphasize on the use of the computer in pure maths. 
I have 3 questions about all this and I don't find references:
1) I know how to compute plethysm in terms of power sum symmetric functions, but why is it defined this way?
2) What is known or conjectured about operations on plethysms that are Schur-positive (i.e. having only nonnegative coefficients when expressed in terms of Schur functions), like the Foulkes' conjecure, that said that $\forall a,b \in \mathbb{N}, \ a \leq b, \ h_b[h_a] - h_a[h_b]$ is Schur-positive, where $h_n = s_{(n)}$, the schur function indexed with only one part?
3) Where can I find clear versions (and with the use of modern notations) of the Thrall's proofs of $h_2[h_n]$, $h_n[h_2]$ and $h_3[h_n]$?
Thanks in advance for short explanations or for references!
 A: (Old question, but maybe future searchers will benefit from an answer.)


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*This is motivated pretty well around (A2.159) in the second appendix to Stanley's Enumerative Combinatorics, volume 2. If you're comfortable with $\mathrm{GL}(V)$-representation theory, you can say it follows more or less immediately from computing trace characters of the composite using weight space decompositions of each factor. A minor wrinkle is that $p_n(x_1, x_2, \ldots)$ is a virtual character for $n \geq 2$, so $p_n[p_m]$ can't literally be defined in this way. However, the "monomial substitution" formula is completely motivated in this way and more generally obviously works for $f[g]$ whenever $f$ is symmetric and $g$ is a sum of monomials, so it's not a real issue.

*This seems too vague to really answer. The marquee problem in the area is of course to give combinatorial interpretations for the non-negative coefficients $g_{\mu,\nu}^\lambda$ in $s_\mu[s_\nu] = \sum_\lambda g_{\mu,\nu}^\lambda s_\lambda$. These are only known to be non-negative from representation theory. Conventional wisdom strongly suggests this problem is much too hard to solve in general. You mentioned Foulkes' conjecture, which is known to be true in some small cases and in a limiting case that's easy to look up....

*The standard modern source for these is Macdonald's Symmetric Functions and Hall Polynomials book, Examples I.8.6,9 pp.138-141. I glanced back at Thrall's 1942 paper (e.g. around Theorem III) and the proof techniques look very different. I don't know of a modern exposition of Thrall's proof; I'd be a bit surprised if it existed.
